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Chapter 17: Problem 64

Show that the coefficient of volume expansion of an ideal gas at constant pressure is the reciprocal of its kelvin temperature.

Short Answer

Expert verified
The coefficient of volume expansion for an ideal gas at constant pressure is the reciprocal of its kelvin temperature.

Step by step solution

01

Write down the ideal gas law

The ideal gas law is given by \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin.
02

Derive the formula for volume expansion

The coefficient of volume expansion \(\beta\) is the ratio of change in volume \(\Delta V\) to the initial volume \(V\) and the change in temperature \(\Delta T\): \(\beta = \frac{\Delta V}{V \Delta T}\).
03

Apply the ideal gas law

From the ideal gas law keeping pressure constant, we can say that \(\Delta V = nR \Delta T\). Substituting this in the above equation, we get \(\beta = \frac{nR \Delta T}{V \Delta T}\). The \(\Delta T\) terms cancel each other, yielding \(\beta = \frac{nR}{V}\).
04

Substitute the ideal gas law into the formula for 尾

Using the ideal gas law again, we substitute \(PV = nRT\) into the Formula for 尾, replacing V with \(\frac{nRT}{P}\), giving \(\beta = \frac{nR}{\frac{nRT}{P}} = \frac{P}{nRT}\).
05

Simplify

Further simplification provides \(\beta = \frac{P}{nRT} = \frac{1}{T}\).

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(a) If \(2.0 \mathrm{~mol}\) of an ideal gas are initially at temperature \(250 \mathrm{~K}\) and pressure \(1.5 \mathrm{~atm}\), what's the gas volume? (b) The pressure is now increased to \(4.0\) atm, and the gas volume drops to half its initial value. What's the new temperature?
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