A car tire contains 0.0380 m^3 of air at a pressure of 2.20 ×10^5 N / m^2 (about 32 psi ). How m

step 1 This is problem 10, chapter 15. And so our ideal gas equation is PV equals NRT, as you know. And

A car tire contains 0.0380 m^3 of air at a pressure of 2.20...

A car tire contains $0.0380 \mathrm{m}^{3}$ of air at a pressure of $2.20 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}$ (about $32 \mathrm{psi}$ ). How much more internal energy does this gas have than the same volume has at zero gauge pressure (which is equivalent to normal atmospheric pressure)?

Key Concepts

Internal Energy of an Ideal Gas

This concept pertains to the microscopic energy stored in an ideal gas due to the kinetic energy of its molecules. For an ideal gas, the internal energy depends solely on the temperature and the degrees of freedom (such as translational, rotational, and possibly vibrational motions) of the molecules, and is independent of the gas’s pressure or volume as long as the temperature remains constant.

Ideal Gas Law

This law, expressed as PV = nRT, relates the pressure, volume, number of moles, temperature, and the universal gas constant for an ideal gas. It provides a framework for understanding how changes in one state variable (such as pressure) affect others (like temperature or the number of moles) and is crucial in determining the number of particles present in a given volume at a specified pressure and temperature.

Gauge Pressure versus Absolute Pressure

Gauge pressure is the pressure measured relative to the ambient atmospheric pressure, while absolute pressure is measured relative to zero pressure (a perfect vacuum). Recognizing this distinction is important in thermodynamic calculations because many relationships, including the ideal gas law, require the use of absolute pressure to accurately determine parameters such as the number of moles and the corresponding internal energy.

Degrees of Freedom

In thermodynamics, degrees of freedom refer to the independent ways in which a molecule can store energy, including movements such as translation, rotation, and vibration. The number of degrees of freedom directly influences the internal energy through the equipartition theorem, where each degree of freedom contributes a specific amount of energy per molecule at a given temperature.

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