The Amount Of Radioactive Element Remaining, R R R , In A 100 Mg 100 \, \text{mg} 100 Mg Sample After D D D Days Is Represented Using The Equation R = 100 ( 1 2 ) D 5 R=100\left(\frac{1}{2}\right)^{\frac{d}{5}} R = 100 ( 2 1 ​ ) 5 D ​ . What Is The Daily Percent Of

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Understanding Radioactive Decay

Radioactive decay is a process in which unstable atomic nuclei lose energy through the emission of radiation. This process is characterized by a half-life, which is the time required for half of the radioactive material to decay. The half-life is a constant for each specific radioactive isotope and is typically measured in units of time, such as years or days.

The Half-Life Equation

The equation for radioactive decay is given by:

r=100(12)d5r=100\left(\frac{1}{2}\right)^{\frac{d}{5}}

where rr is the amount of radioactive element remaining in a 100 mg100 \, \text{mg} sample after dd days.

Interpreting the Equation

In this equation, the base of the exponent is 12\frac{1}{2}, which represents the fact that the amount of radioactive material decreases by half every 5 days. This is known as the half-life of the radioactive isotope.

Calculating the Daily Percent of Radioactive Decay

To calculate the daily percent of radioactive decay, we need to find the ratio of the amount of radioactive material remaining after one day to the initial amount. This can be done by substituting d=1d=1 into the equation:

r=100(12)15r=100\left(\frac{1}{2}\right)^{\frac{1}{5}}

Simplifying the Equation

To simplify the equation, we can use the fact that (12)15\left(\frac{1}{2}\right)^{\frac{1}{5}} is equal to (12)15\left(\frac{1}{2}\right)^{\frac{1}{5}}. This can be rewritten as:

r=100(12)15r=100\left(\frac{1}{2}\right)^{\frac{1}{5}}

Evaluating the Expression

To evaluate the expression, we can use a calculator or a computer program to find the value of (12)15\left(\frac{1}{2}\right)^{\frac{1}{5}}. This is approximately equal to 0.87060.8706.

Calculating the Daily Percent of Radioactive Decay

Now that we have the value of (12)15\left(\frac{1}{2}\right)^{\frac{1}{5}}, we can calculate the daily percent of radioactive decay by dividing the amount of radioactive material remaining after one day by the initial amount:

Daily percent of radioactive decay=100(12)15100×100%\text{Daily percent of radioactive decay} = \frac{100\left(\frac{1}{2}\right)^{\frac{1}{5}}}{100} \times 100\%

Simplifying the Expression

To simplify the expression, we can cancel out the 100100 in the numerator and denominator:

Daily percent of radioactive decay=(12)15×100%\text{Daily percent of radioactive decay} = \left(\frac{1}{2}\right)^{\frac{1}{5}} \times 100\%

Evaluating the Expression

To evaluate the expression, we can multiply the value of (12)15\left(\frac{1}{2}\right)^{\frac{1}{5}} by 100%100\%:

Daily percent of radioactive decay=0.8706×100%=87.06%\text{Daily percent of radioactive decay} = 0.8706 \times 100\% = 87.06\%

Conclusion

In this article, we have used the half-life equation to calculate the daily percent of radioactive decay for a 100 mg100 \, \text{mg} sample of radioactive material. We have shown that the daily percent of radioactive decay is approximately 87.06%87.06\%.

References

Appendix

Derivation of the Half-Life Equation

The half-life equation can be derived from the following assumptions:

  • The amount of radioactive material remaining after a certain time is proportional to the initial amount.
  • The rate of radioactive decay is constant.

Using these assumptions, we can derive the half-life equation as follows:

Let N(t)N(t) be the amount of radioactive material remaining after time tt. Then, we can write:

N(t)=N0e−λtN(t) = N_0 e^{-\lambda t}

where N0N_0 is the initial amount of radioactive material, λ\lambda is the decay constant, and tt is time.

The half-life equation can be obtained by substituting t=t1/2t = t_{1/2} into the above equation, where t1/2t_{1/2} is the half-life:

N(t1/2)=N0e−λt1/2N(t_{1/2}) = N_0 e^{-\lambda t_{1/2}}

Since N(t1/2)=12N0N(t_{1/2}) = \frac{1}{2} N_0, we can write:

12N0=N0e−λt1/2\frac{1}{2} N_0 = N_0 e^{-\lambda t_{1/2}}

Dividing both sides by N0N_0, we get:

12=e−λt1/2\frac{1}{2} = e^{-\lambda t_{1/2}}

Taking the natural logarithm of both sides, we get:

ln⁡12=−λt1/2\ln \frac{1}{2} = -\lambda t_{1/2}

Solving for λ\lambda, we get:

λ=−ln⁡12t1/2\lambda = -\frac{\ln \frac{1}{2}}{t_{1/2}}

Substituting this expression for λ\lambda into the half-life equation, we get:

N(t)=N0e−ln⁡12t1/2tN(t) = N_0 e^{-\frac{\ln \frac{1}{2}}{t_{1/2}} t}

Simplifying this expression, we get:

N(t)=N0(12)tt1/2N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}

This is the half-life equation.

Derivation of the Daily Percent of Radioactive Decay

To derive the daily percent of radioactive decay, we can use the half-life equation:

N(t)=N0(12)tt1/2N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}

Substituting t=1t = 1 day into this equation, we get:

N(1)=N0(12)1t1/2N(1) = N_0 \left(\frac{1}{2}\right)^{\frac{1}{t_{1/2}}}

Since t1/2=5t_{1/2} = 5 days, we can write:

N(1)=N0(12)15N(1) = N_0 \left(\frac{1}{2}\right)^{\frac{1}{5}}

Dividing both sides by N0N_0, we get:

N(1)N0=(12)15\frac{N(1)}{N_0} = \left(\frac{1}{2}\right)^{\frac{1}{5}}

Multiplying both sides by 100%100\%, we get:

Daily percent of radioactive decay=(12)15×100%\text{Daily percent of radioactive decay} = \left(\frac{1}{2}\right)^{\frac{1}{5}} \times 100\%

Evaluating this expression, we get:

\text{Daily percent of radioactive decay} = 0.8706 \times 100\% = 87.06\%$<br/> **The Half-Life Equation and Daily Radioactive Decay: Q&A** =====================================================

Q: What is the half-life equation?

A: The half-life equation is a mathematical formula that describes the rate of radioactive decay of a substance. It is given by:

r=100(12)d5</span></p><p>where<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotationencoding="application/x−tex">r</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span></span></span></span>istheamountofradioactiveelementremainingina<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotationencoding="application/x−tex">100 mg</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8389em;vertical−align:−0.1944em;"></span><spanclass="mord">100</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordtext"><spanclass="mord">mg</span></span></span></span></span>sampleafter<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotationencoding="application/x−tex">d</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mordmathnormal">d</span></span></span></span>days.</p><h2><strong>Q:Whatisthesignificanceofthehalf−lifeequation?</strong></h2><p>A:Thehalf−lifeequationissignificantbecauseitallowsustocalculatetheamountofradioactivematerialremaininginasampleafteracertainperiodoftime.Thisisusefulinavarietyofapplications,includingnuclearmedicine,geology,andenvironmentalscience.</p><h2><strong>Q:HowdoIcalculatethedailypercentofradioactivedecay?</strong></h2><p>A:Tocalculatethedailypercentofradioactivedecay,youcanusethefollowingformula:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mtext>Daily percent of radioactive decay</mtext><mo>=</mo><msup><mrow><mofence="true">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mofence="true">)</mo></mrow><mfrac><mn>1</mn><mn>5</mn></mfrac></msup><mo>×</mo><mn>100</mn><mimathvariant="normal"></annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordtext"><spanclass="mord">Daily percent of radioactive decay</span></span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:2.744em;vertical−align:−0.95em;"></span><spanclass="minner"><spanclass="minner"><spanclass="mopendelimcenter"style="top:0em;"><spanclass="delimsizingsize3">(</span></span><spanclass="mord"><spanclass="mopennulldelimiter"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:1.3214em;"><spanstyle="top:−2.314em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">2</span></span></span><spanstyle="top:−3.23em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−line"style="border−bottom−width:0.04em;"></span></span><spanstyle="top:−3.677em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="mord"><spanclass="mord">1</span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.686em;"><span></span></span></span></span></span><spanclass="mclosenulldelimiter"></span></span><spanclass="mclosedelimcenter"style="top:0em;"><spanclass="delimsizingsize3">)</span></span></span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:1.7939em;"><spanstyle="top:−4.2029em;margin−right:0.05em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight"><spanclass="mopennulldelimitersizingreset−size3size6"></span><spanclass="mfrac"><spanclass="vlist−tvlist−t2"><spanclass="vlist−r"><spanclass="vlist"style="height:0.8443em;"><spanstyle="top:−2.656em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmtight">5</span></span></span></span><spanstyle="top:−3.2255em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="frac−linemtight"style="border−bottom−width:0.049em;"></span></span><spanstyle="top:−3.384em;"><spanclass="pstrut"style="height:3em;"></span><spanclass="sizingreset−size3size1mtight"><spanclass="mordmtight"><spanclass="mordmtight">1</span></span></span></span></span><spanclass="vlist−s">​</span></span><spanclass="vlist−r"><spanclass="vlist"style="height:0.344em;"><span></span></span></span></span></span><spanclass="mclosenulldelimitersizingreset−size3size6"></span></span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">×</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.8056em;vertical−align:−0.0556em;"></span><spanclass="mord">100<p>Thisformulaisderivedfromthehalf−lifeequationandtakesintoaccountthefactthattheamountofradioactivematerialdecreasesbyhalfevery5days.</p><h2><strong>Q:Whatisthedailypercentofradioactivedecayfora<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotationencoding="application/x−tex">100 mg</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8389em;vertical−align:−0.1944em;"></span><spanclass="mord">100</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordtext"><spanclass="mord">mg</span></span></span></span></span>sampleofradioactivematerial?</strong></h2><p>A:Thedailypercentofradioactivedecayfora<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotationencoding="application/x−tex">100 mg</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8389em;vertical−align:−0.1944em;"></span><spanclass="mord">100</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordtext"><spanclass="mord">mg</span></span></span></span></span>sampleofradioactivematerialisapproximately<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>87.06</mn><mimathvariant="normal"><h2><strong>Q:Howdoesthehalf−lifeequationrelatetotheconceptofhalf−life?</strong></h2><p>A:Thehalf−lifeequationiscloselyrelatedtotheconceptofhalf−life.Thehalf−lifeofaradioactivesubstanceisthetimeittakesforhalfoftheoriginalamountofthesubstancetodecay.Thehalf−lifeequationtakesintoaccountthisconceptandallowsustocalculatetheamountofradioactivematerialremaininginasampleafteracertainperiodoftime.</p><h2><strong>Q:CanIusethehalf−lifeequationtocalculatetheamountofradioactivematerialremaininginasampleafteracertainperiodoftime?</strong></h2><p>A:Yes,youcanusethehalf−lifeequationtocalculatetheamountofradioactivematerialremaininginasampleafteracertainperiodoftime.Simplysubstitutethedesiredtimeintotheequationandsolvefortheamountofradioactivematerialremaining.</p><h2><strong>Q:Whataresomecommonapplicationsofthehalf−lifeequation?</strong></h2><p>A:Thehalf−lifeequationhasavarietyofapplicationsinfieldssuchasnuclearmedicine,geology,andenvironmentalscience.Somecommonapplicationsinclude:</p><ul><li>Calculatingtheamountofradioactivematerialremaininginasampleafteracertainperiodoftime</li><li>Determiningthehalf−lifeofaradioactivesubstance</li><li>Modelingthedecayofradioactivematerialsintheenvironment</li><li>Calculatingthedoseofradiationreceivedbyapatientinanuclearmedicineprocedure</li></ul><h2><strong>Q:CanIusethehalf−lifeequationtocalculatethedailypercentofradioactivedecayforasampleofradioactivematerialwithadifferentinitialamount?</strong></h2><p>A:Yes,youcanusethehalf−lifeequationtocalculatethedailypercentofradioactivedecayforasampleofradioactivematerialwithadifferentinitialamount.Simplysubstitutethedesiredinitialamountintotheequationandsolveforthedailypercentofradioactivedecay.</p><h2><strong>Q:Whataresomelimitationsofthehalf−lifeequation?</strong></h2><p>A:Thehalf−lifeequationhasseverallimitations,including:</p><ul><li>Itassumesthattherateofradioactivedecayisconstantovertime</li><li>Itassumesthattheamountofradioactivematerialremaininginasampleisdirectlyproportionaltotheinitialamount</li><li>Itdoesnottakeintoaccounttheeffectsofexternalfactors,suchastemperatureandpressure,ontherateofradioactivedecay</li></ul><h2><strong>Q:CanIusethehalf−lifeequationtocalculatetheamountofradioactivematerialremaininginasampleafteracertainperiodoftimeifthesampleisnota<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotationencoding="application/x−tex">100 mg</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8389em;vertical−align:−0.1944em;"></span><spanclass="mord">100</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordtext"><spanclass="mord">mg</span></span></span></span></span>sample?</strong></h2><p>A:Yes,youcanusethehalf−lifeequationtocalculatetheamountofradioactivematerialremaininginasampleafteracertainperiodoftimeifthesampleisnota<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotationencoding="application/x−tex">100 mg</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.8389em;vertical−align:−0.1944em;"></span><spanclass="mord">100</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordtext"><spanclass="mord">mg</span></span></span></span></span>sample.Simplysubstitutethedesiredinitialamountintotheequationandsolvefortheamountofradioactivematerialremaining.</p><h2><strong>Q:Whataresomecommonmistakestoavoidwhenusingthehalf−lifeequation?</strong></h2><p>A:Somecommonmistakestoavoidwhenusingthehalf−lifeequationinclude:</p><ul><li>Failingtoaccountfortheeffectsofexternalfactors,suchastemperatureandpressure,ontherateofradioactivedecay</li><li>Usingthehalf−lifeequationtocalculatetheamountofradioactivematerialremaininginasampleafteracertainperiodoftimewithouttakingintoaccounttheinitialamountofthesample</li><li>Failingtousethecorrectunitswhensubstitutingvaluesintotheequation</li></ul><h2><strong>Q:CanIusethehalf−lifeequationtocalculatethedailypercentofradioactivedecayforasampleofradioactivematerialwithadifferenthalf−life?</strong></h2><p>A:Yes,youcanusethehalf−lifeequationtocalculatethedailypercentofradioactivedecayforasampleofradioactivematerialwithadifferenthalf−life.Simplysubstitutethedesiredhalf−lifeintotheequationandsolveforthedailypercentofradioactivedecay.</p><h2><strong>Q:Whataresomeresourcesavailableforlearningmoreaboutthehalf−lifeequationanditsapplications?</strong></h2><p>A:Someresourcesavailableforlearningmoreaboutthehalf−lifeequationanditsapplicationsinclude:</p><ul><li>Textbooksonnuclearphysicsandradiationsafety</li><li>Onlinecoursesandtutorialsonnuclearphysicsandradiationsafety</li><li>Professionalorganizations,suchastheAmericanNuclearSocietyandtheInternationalAtomicEnergyAgency</li><li>Researcharticlesandpapersonthehalf−lifeequationanditsapplications</li></ul>r=100\left(\frac{1}{2}\right)^{\frac{d}{5}} </span></p> <p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> is the amount of radioactive element remaining in a <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotation encoding="application/x-tex">100 \, \text{mg}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">mg</span></span></span></span></span> sample after <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> days.</p> <h2><strong>Q: What is the significance of the half-life equation?</strong></h2> <p>A: The half-life equation is significant because it allows us to calculate the amount of radioactive material remaining in a sample after a certain period of time. This is useful in a variety of applications, including nuclear medicine, geology, and environmental science.</p> <h2><strong>Q: How do I calculate the daily percent of radioactive decay?</strong></h2> <p>A: To calculate the daily percent of radioactive decay, you can use the following formula:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>Daily percent of radioactive decay</mtext><mo>=</mo><msup><mrow><mo fence="true">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo fence="true">)</mo></mrow><mfrac><mn>1</mn><mn>5</mn></mfrac></msup><mo>×</mo><mn>100</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\text{Daily percent of radioactive decay} = \left(\frac{1}{2}\right)^{\frac{1}{5}} \times 100\% </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Daily percent of radioactive decay</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.744em;vertical-align:-0.95em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.7939em;"><span style="top:-4.2029em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">100%</span></span></span></span></span></p> <p>This formula is derived from the half-life equation and takes into account the fact that the amount of radioactive material decreases by half every 5 days.</p> <h2><strong>Q: What is the daily percent of radioactive decay for a <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotation encoding="application/x-tex">100 \, \text{mg}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">mg</span></span></span></span></span> sample of radioactive material?</strong></h2> <p>A: The daily percent of radioactive decay for a <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotation encoding="application/x-tex">100 \, \text{mg}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">mg</span></span></span></span></span> sample of radioactive material is approximately <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>87.06</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">87.06\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">87.06%</span></span></span></span>. This means that after one day, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>87.06</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">87.06\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8056em;vertical-align:-0.0556em;"></span><span class="mord">87.06%</span></span></span></span> of the original amount of radioactive material will remain.</p> <h2><strong>Q: How does the half-life equation relate to the concept of half-life?</strong></h2> <p>A: The half-life equation is closely related to the concept of half-life. The half-life of a radioactive substance is the time it takes for half of the original amount of the substance to decay. The half-life equation takes into account this concept and allows us to calculate the amount of radioactive material remaining in a sample after a certain period of time.</p> <h2><strong>Q: Can I use the half-life equation to calculate the amount of radioactive material remaining in a sample after a certain period of time?</strong></h2> <p>A: Yes, you can use the half-life equation to calculate the amount of radioactive material remaining in a sample after a certain period of time. Simply substitute the desired time into the equation and solve for the amount of radioactive material remaining.</p> <h2><strong>Q: What are some common applications of the half-life equation?</strong></h2> <p>A: The half-life equation has a variety of applications in fields such as nuclear medicine, geology, and environmental science. Some common applications include:</p> <ul> <li>Calculating the amount of radioactive material remaining in a sample after a certain period of time</li> <li>Determining the half-life of a radioactive substance</li> <li>Modeling the decay of radioactive materials in the environment</li> <li>Calculating the dose of radiation received by a patient in a nuclear medicine procedure</li> </ul> <h2><strong>Q: Can I use the half-life equation to calculate the daily percent of radioactive decay for a sample of radioactive material with a different initial amount?</strong></h2> <p>A: Yes, you can use the half-life equation to calculate the daily percent of radioactive decay for a sample of radioactive material with a different initial amount. Simply substitute the desired initial amount into the equation and solve for the daily percent of radioactive decay.</p> <h2><strong>Q: What are some limitations of the half-life equation?</strong></h2> <p>A: The half-life equation has several limitations, including:</p> <ul> <li>It assumes that the rate of radioactive decay is constant over time</li> <li>It assumes that the amount of radioactive material remaining in a sample is directly proportional to the initial amount</li> <li>It does not take into account the effects of external factors, such as temperature and pressure, on the rate of radioactive decay</li> </ul> <h2><strong>Q: Can I use the half-life equation to calculate the amount of radioactive material remaining in a sample after a certain period of time if the sample is not a <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotation encoding="application/x-tex">100 \, \text{mg}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">mg</span></span></span></span></span> sample?</strong></h2> <p>A: Yes, you can use the half-life equation to calculate the amount of radioactive material remaining in a sample after a certain period of time if the sample is not a <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext> </mtext><mtext>mg</mtext></mrow><annotation encoding="application/x-tex">100 \, \text{mg}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">mg</span></span></span></span></span> sample. Simply substitute the desired initial amount into the equation and solve for the amount of radioactive material remaining.</p> <h2><strong>Q: What are some common mistakes to avoid when using the half-life equation?</strong></h2> <p>A: Some common mistakes to avoid when using the half-life equation include:</p> <ul> <li>Failing to account for the effects of external factors, such as temperature and pressure, on the rate of radioactive decay</li> <li>Using the half-life equation to calculate the amount of radioactive material remaining in a sample after a certain period of time without taking into account the initial amount of the sample</li> <li>Failing to use the correct units when substituting values into the equation</li> </ul> <h2><strong>Q: Can I use the half-life equation to calculate the daily percent of radioactive decay for a sample of radioactive material with a different half-life?</strong></h2> <p>A: Yes, you can use the half-life equation to calculate the daily percent of radioactive decay for a sample of radioactive material with a different half-life. Simply substitute the desired half-life into the equation and solve for the daily percent of radioactive decay.</p> <h2><strong>Q: What are some resources available for learning more about the half-life equation and its applications?</strong></h2> <p>A: Some resources available for learning more about the half-life equation and its applications include:</p> <ul> <li>Textbooks on nuclear physics and radiation safety</li> <li>Online courses and tutorials on nuclear physics and radiation safety</li> <li>Professional organizations, such as the American Nuclear Society and the International Atomic Energy Agency</li> <li>Research articles and papers on the half-life equation and its applications</li> </ul>