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Chapter 13: Problem 59

Consider the expansion of an ideal gas at constant pressure. The initial temperature is \(T_{0}\) and the initial volume is \(V_{0} .\) (a) Show that \(\Delta V / V_{0}=\beta \Delta T,\) where \(\beta=1 / T_{0}\) (b) Compare the coefficient of volume expansion \(\beta\) for an ideal gas at \(20^{\circ} \mathrm{C}\) to the values for liquids and gases listed in Table 13.3

Short Answer

Expert verified
#Answer#: The relationship between the change in volume and the change in temperature at constant pressure for an ideal gas is given by the equation: \(\frac{\Delta V}{V_{0}} = \beta \Delta T\), where \(\beta = \frac{1}{T_{0}}\). The coefficient of volume expansion for an ideal gas at \(20^{\circ} \mathrm{C} = 293 \mathrm{K}\) is \(\beta = \frac{1}{293 \mathrm{K}}\). Comparison to the values for liquids and gases listed in Table 13.3 (not provided) will show any similarities or differences between the volume expansion coefficients for an ideal gas, liquids, and other gases at the given temperature.

Step by step solution

01

Understand the Ideal Gas Law

The Ideal Gas Law states that \(PV = nRT\), where P is pressure, V is volume, n is the amount of substance (in moles), R is the ideal gas constant, and T is the temperature in Kelvin. Since pressure is constant in this problem, we can rewrite the Ideal Gas Law as \(V = nRT/P = constant * T\). To find the relationship between the change in volume and the change in temperature, we will need to differentiate this equation with respect to T.
02

Differentiate the equation with respect to T

Begin by differentiating \(V = constant * T\) with respect to T: \[ \frac{dV}{dT} = constant \] This implies that the ratio of the change in volume to the change in temperature is constant. Now we can rewrite this expression in terms of initial conditions and volume:
03

Rewrite the equation in terms of initial conditions and volume change

First, find the change in volume: \[ \Delta V = V - V_{0} \] Next, find the change in temperature: \[ \Delta T = T - T_{0} \] Now, we can rewrite the expression \(\frac{dV}{dT} = constant\) as: \[\frac{\Delta V}{\Delta T} = \frac{V - V_{0}}{T - T_{0}} = constant \] Since \(V = constant * T\), and initial volume \(V_{0} = constant * T_{0}\), we can write: \[\frac{\Delta V}{V_{0}} = \frac{V - V_{0}}{constant * T_{0}} = \frac{T - T_{0}}{T_{0}}\] Express \(\beta\) in terms of \(T_{0}\): \[ \beta = \frac{1}{T_{0}} \]
04

Derive the relationship between \(\Delta V, V_{0}, \beta,\) and \(\Delta T\)

Now we can substitute \(\beta\) into the equation to obtain the desired relationship: \[\frac{\Delta V}{V_{0}} = \beta \Delta T\]
05

Compare the coefficient of volume expansion for an ideal gas at \(20^{\circ} \mathrm{C}\) with the values for liquids and gases

Assuming the available table for volume expansion is similar to ones found in standard physics texts, you can compare the volume expansion coefficient of an ideal gas at \(20^{\circ} \mathrm{C} = 293 \mathrm{K}\) to the values for liquids and solids (which we do not have). To determine the coefficient of volume expansion for an ideal gas at \(20^{\circ} \mathrm{C}\), you can use the formula for \(\beta\): \[ \beta = \frac{1}{T_{0}} = \frac{1}{293 \mathrm{K}}\] Now you can compare the coefficient of volume expansion for the ideal gas at \(20^{\circ} \mathrm{C}\) to the values for liquids and gases found in the provided table and discuss any similarities or differences between the values.

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