How Many Moles Of Air Are There In A Bicycle Tire With A Volume Of 2.36 L If It Has An Internal Pressure Of 6.8 Atm At 17.0 ∘ C 17.0^{\circ} C 17. 0 ∘ C ?

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Understanding the Problem

In this problem, we are tasked with determining the number of moles of air present in a bicycle tire with a specific volume and internal pressure. To solve this problem, we will use the ideal gas law, which relates the pressure, volume, and temperature of a gas.

The Ideal Gas Law

The ideal gas law is given by the equation:

PV = nRT

where:

  • P is the pressure of the gas (in atm)
  • V is the volume of the gas (in L)
  • n is the number of moles of the gas
  • R is the gas constant (8.314 L atm/mol K)
  • T is the temperature of the gas (in K)

Given Values

  • V = 2.36 L
  • P = 6.8 atm
  • T = 17.0°C = 290 K

Converting Temperature to Kelvin

The temperature given in the problem is in Celsius, but the ideal gas law requires the temperature to be in Kelvin. To convert the temperature from Celsius to Kelvin, we use the following equation:

T (K) = T (°C) + 273.15

Plugging in the given temperature, we get:

T (K) = 17.0°C + 273.15 = 290 K

Solving for n

Now that we have all the given values, we can plug them into the ideal gas law equation and solve for n:

PV = nRT

Rearranging the equation to solve for n, we get:

n = PV / RT

Plugging in the given values, we get:

n = (6.8 atm)(2.36 L) / (8.314 L atm/mol K)(290 K)

n = 0.061 mol

Conclusion

Therefore, there are approximately 0.061 moles of air present in a bicycle tire with a volume of 2.36 L and an internal pressure of 6.8 atm at 17.0°C.

Real-World Applications

This problem has real-world applications in various fields, such as:

  • Aerospace Engineering: Understanding the properties of gases is crucial in the design of aircraft and spacecraft.
  • Chemical Engineering: The ideal gas law is used to design and optimize chemical processes, such as the production of fuels and chemicals.
  • Environmental Science: The ideal gas law is used to study the behavior of gases in the atmosphere and to understand the effects of climate change.

Limitations of the Ideal Gas Law

The ideal gas law is a simplified model that assumes ideal behavior, which is not always the case in real-world situations. The ideal gas law assumes that the gas molecules are point particles with no volume, that there are no intermolecular forces, and that the gas is in thermal equilibrium. However, in reality, gas molecules have volume, there are intermolecular forces, and the gas may not be in thermal equilibrium. Therefore, the ideal gas law is only an approximation and should be used with caution.

Future Research Directions

Future research directions in this area include:

  • Developing more accurate models: Developing more accurate models that take into account the complexities of real-world gases.
  • Improving the ideal gas law: Improving the ideal gas law by incorporating more realistic assumptions and parameters.
  • Applying the ideal gas law to real-world problems: Applying the ideal gas law to real-world problems, such as the design of aircraft and spacecraft, and the study of the behavior of gases in the atmosphere.

References

  • Hall, K. R. (2017). Chemical Thermodynamics: Principles and Applications.
  • Levine, I. N. (2017). Physical Chemistry.
  • McQuarrie, D. A. (2017). Statistical Mechanics.

Q: What is the ideal gas law?

A: The ideal gas law is a mathematical equation that relates the pressure, volume, and temperature of a gas. It is given by the equation PV = nRT, where P is the pressure of the gas, V is the volume of the gas, n is the number of moles of the gas, R is the gas constant, and T is the temperature of the gas.

Q: What are the assumptions of the ideal gas law?

A: The ideal gas law assumes that the gas molecules are point particles with no volume, that there are no intermolecular forces, and that the gas is in thermal equilibrium. However, in reality, gas molecules have volume, there are intermolecular forces, and the gas may not be in thermal equilibrium.

Q: What are the limitations of the ideal gas law?

A: The ideal gas law is a simplified model that is only an approximation of real-world gases. It does not take into account the complexities of real-world gases, such as the effects of intermolecular forces and the non-ideal behavior of gases at high pressures and low temperatures.

Q: How is the ideal gas law used in real-world applications?

A: The ideal gas law is used in a wide range of real-world applications, including:

  • Aerospace Engineering: The ideal gas law is used to design and optimize aircraft and spacecraft.
  • Chemical Engineering: The ideal gas law is used to design and optimize chemical processes, such as the production of fuels and chemicals.
  • Environmental Science: The ideal gas law is used to study the behavior of gases in the atmosphere and to understand the effects of climate change.

Q: What are some common mistakes to avoid when using the ideal gas law?

A: Some common mistakes to avoid when using the ideal gas law include:

  • Not converting temperature to Kelvin: The ideal gas law requires the temperature to be in Kelvin, not Celsius.
  • Not using the correct units: The ideal gas law requires the pressure and volume to be in the same units, such as atmospheres and liters.
  • Not taking into account the limitations of the ideal gas law: The ideal gas law is only an approximation of real-world gases, and should not be used in situations where the gas is not ideal.

Q: What are some alternative models to the ideal gas law?

A: Some alternative models to the ideal gas law include:

  • The van der Waals equation: This equation takes into account the effects of intermolecular forces and the non-ideal behavior of gases at high pressures and low temperatures.
  • The Redlich-Kwong equation: This equation is a more accurate model of real-world gases than the ideal gas law, and is often used in chemical engineering applications.
  • The Peng-Robinson equation: This equation is a more accurate model of real-world gases than the ideal gas law, and is often used in chemical engineering applications.

Q: What are some real-world examples of the ideal gas law in action?

A: Some real-world examples of the ideal gas law in action include:

  • Scuba diving: The ideal gas law is used to calculate the pressure of the air in a scuba tank, and to determine the safe depth for a dive.
  • Aircraft design: The ideal gas law is used to design and optimize aircraft, and to calculate the pressure and temperature of the air in the aircraft.
  • Chemical processing: The ideal gas law is used to design and optimize chemical processes, such as the production of fuels and chemicals.

Q: What are some future research directions in the field of ideal gas law?

A: Some future research directions in the field of ideal gas law include:

  • Developing more accurate models: Developing more accurate models that take into account the complexities of real-world gases.
  • Improving the ideal gas law: Improving the ideal gas law by incorporating more realistic assumptions and parameters.
  • Applying the ideal gas law to real-world problems: Applying the ideal gas law to real-world problems, such as the design of aircraft and spacecraft, and the study of the behavior of gases in the atmosphere.