A Gas At 49.3 ∘ C 49.3^{\circ} C 49. 3 ∘ C And 893 Mm Hg Experiences A Temperature Change And Ends Up With A Pressure Of 778 Mm Hg. What Is The New Temperature Of The Gas In Degrees Celsius? T = [ ? ] ∘ C T = [?]^{\circ} C T = [ ? ] ∘ C Hint: Be Sure To Watch Your Temperature Units!
Understanding the Ideal Gas Law
The ideal gas law is a fundamental concept in chemistry that describes the behavior of gases under various conditions. It is expressed by the equation:
PV = nRT
Where:
- P is the pressure of the gas in atmospheres (atm)
- V is the volume of the gas in liters (L)
- n is the number of moles of the gas
- R is the gas constant, which is equal to 0.08206 L atm/mol K
- T is the temperature of the gas in Kelvin (K)
Converting Between Celsius and Kelvin
To work with the ideal gas law, we need to convert the temperature from Celsius to Kelvin. The formula for this conversion is:
T (K) = T (°C) + 273.15
Where T (K) is the temperature in Kelvin and T (°C) is the temperature in Celsius.
Applying the Ideal Gas Law to the Problem
We are given a gas at an initial temperature of 49.3°C and an initial pressure of 893 mm Hg. The gas undergoes a temperature change and ends up with a final pressure of 778 mm Hg. We need to find the new temperature of the gas in degrees Celsius.
First, we need to convert the initial and final pressures from mm Hg to atm. We know that 1 atm is equal to 760 mm Hg, so we can make the following conversions:
P1 (atm) = 893 mm Hg / 760 mm Hg/atm = 1.175 atm
P2 (atm) = 778 mm Hg / 760 mm Hg/atm = 1.023 atm
Next, we can use the ideal gas law to relate the initial and final states of the gas. Since the number of moles of the gas remains constant, we can write:
P1V1/T1 = P2V2/T2
We can rearrange this equation to solve for the final temperature:
T2 = P2V2T1 / P1V1
However, we don't know the volume of the gas, so we can't use this equation directly. Instead, we can use the fact that the volume of the gas remains constant to write:
P1/T1 = P2/T2
We can rearrange this equation to solve for the final temperature:
T2 = P2T1 / P1
Now we can plug in the values we know:
T2 = (1.023 atm)(49.3°C + 273.15 K) / (1.175 atm)
T2 = (1.023 atm)(322.45 K) / (1.175 atm)
T2 = 274.5 K
Finally, we can convert the final temperature from Kelvin to Celsius:
T2 (°C) = 274.5 K - 273.15 = 1.35°C
Therefore, the new temperature of the gas is 1.35°C.
Conclusion
In this problem, we used the ideal gas law to relate the initial and final states of a gas undergoing a temperature and pressure change. We were able to find the new temperature of the gas by using the fact that the volume of the gas remains constant. This problem demonstrates the importance of understanding the ideal gas law and how to apply it to real-world problems.
References
- Kittel, C. (2005). Introduction to Solid State Physics. John Wiley & Sons.
- Hall, J. D. (2013). Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics. Cambridge University Press.
Additional Resources
- Ideal Gas Law Calculator: A calculator that can be used to solve problems involving the ideal gas law.
- Thermodynamics Tutorial: A tutorial that provides an introduction to thermodynamics and the ideal gas law.
- Gas Laws: A website that provides information on the gas laws, including the ideal gas law.
A Gas Undergoing Temperature and Pressure Changes =====================================================
Q&A: Understanding the Ideal Gas Law and Temperature Changes
Q: What is the ideal gas law?
A: The ideal gas law is a fundamental concept in chemistry that describes the behavior of gases under various conditions. It is expressed by the equation:
PV = nRT
Where:
- P is the pressure of the gas in atmospheres (atm)
- V is the volume of the gas in liters (L)
- n is the number of moles of the gas
- R is the gas constant, which is equal to 0.08206 L atm/mol K
- T is the temperature of the gas in Kelvin (K)
Q: How do I convert between Celsius and Kelvin?
A: To convert between Celsius and Kelvin, you can use the following formula:
T (K) = T (°C) + 273.15
Where T (K) is the temperature in Kelvin and T (°C) is the temperature in Celsius.
Q: How do I apply the ideal gas law to a problem involving temperature and pressure changes?
A: To apply the ideal gas law to a problem involving temperature and pressure changes, you can use the following steps:
- Convert the initial and final pressures from mm Hg to atm.
- Use the ideal gas law to relate the initial and final states of the gas.
- Solve for the final temperature.
Q: What if I don't know the volume of the gas?
A: If you don't know the volume of the gas, you can use the fact that the volume of the gas remains constant to write:
P1/T1 = P2/T2
You can then rearrange this equation to solve for the final temperature:
T2 = P2T1 / P1
Q: How do I convert the final temperature from Kelvin to Celsius?
A: To convert the final temperature from Kelvin to Celsius, you can use the following formula:
T2 (°C) = T2 (K) - 273.15
Q: What if I want to find the new temperature of the gas in degrees Celsius?
A: To find the new temperature of the gas in degrees Celsius, you can use the following steps:
- Convert the initial temperature from Celsius to Kelvin.
- Use the ideal gas law to relate the initial and final states of the gas.
- Solve for the final temperature in Kelvin.
- Convert the final temperature from Kelvin to Celsius.
Q: What are some common mistakes to avoid when working with the ideal gas law?
A: Some common mistakes to avoid when working with the ideal gas law include:
- Failing to convert between Celsius and Kelvin.
- Failing to convert between mm Hg and atm.
- Failing to use the correct units for the gas constant (R).
- Failing to use the correct units for the temperature (T).
Q: What are some real-world applications of the ideal gas law?
A: Some real-world applications of the ideal gas law include:
- Calculating the pressure of a gas in a container.
- Calculating the volume of a gas in a container.
- Calculating the temperature of a gas in a container.
- Calculating the number of moles of a gas in a container.
Conclusion
In this Q&A article, we have discussed the ideal gas law and how to apply it to problems involving temperature and pressure changes. We have also discussed some common mistakes to avoid and some real-world applications of the ideal gas law. By understanding the ideal gas law and how to apply it, you can solve a wide range of problems in chemistry and other fields.
References
- Kittel, C. (2005). Introduction to Solid State Physics. John Wiley & Sons.
- Hall, J. D. (2013). Thermodynamics: An Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics. Cambridge University Press.
Additional Resources
- Ideal Gas Law Calculator: A calculator that can be used to solve problems involving the ideal gas law.
- Thermodynamics Tutorial: A tutorial that provides an introduction to thermodynamics and the ideal gas law.
- Gas Laws: A website that provides information on the gas laws, including the ideal gas law.