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Chapter 6: Problem 85

A quantity of 0.020 mole of a gas initially at \(0.050 \mathrm{~L}\) and \(20^{\circ} \mathrm{C}\) undergoes a constant-temperature expansion until its volume is \(0.50 \mathrm{~L}\). Calculate the work done (in joules) by the gas if it expands (a) against a vacuum and (b) against a constant pressure of 0.20 atm. (c) If the gas in (b) is allowed to expand unchecked until its pressure is equal to the external pressure, what would its final volume be before it stopped expanding, and what would be the work done?

Short Answer

Expert verified
Based on the above calculations, (a) the work done by the gas when it expands against a vacuum is 0J, (b) the work done by the gas when it expands against a constant external pressure of 0.20 atm can be calculated using the formula \(W = -P * ΔV\), (c) and the final volume when the gas pressure equals the external pressure is equal to the initial volume, and the work done by the gas in this condition can again be evaluated using the same formula for work done.

Step by step solution

01

Calculate Initial Pressure

To begin with, the initial pressure of the gas needs to be calculated. We can use the ideal gas equation for that, which is \(P_i V_i = n R T_i\). Given that \(V_i = 0.050 L\), \(n = 0.020 mol\), \(R = 0.0821 L atm /(mol K)\) (gas constant in desired units) and \(T_i = 20 + 273.15 = 293.15 K (absolute temperature)\), we just need to solve for \(P_i\).
02

Calculate Work Done in Expansion Against a Vacuum

When a gas expands against vacuum (as in case a), there is no external pressure, and since work done by the gas is given by the formula \(W = -P * ΔV\), in this case the work done by the gas would be \(W = -0 * (final volume - initial volume) = 0J\).
03

Calculate Work Done in Expansion Against a Constant External Pressure

In case (b), the gas is expanding against a constant external pressure of \(0.20 atm\). Using the same work done formula \(W = -P * ΔV\), and given that final volume \(V_f = 0.50 L\) and initial volume \(V_i = 0.050 L\), we find the work done by the gas against the constant external pressure.
04

Find Final Volume and Work Done When Gas Pressure Equals External Pressure

In case (c), the gas expands until its pressure equals the external pressure. At this point, we apply the ideal gas law again: \(P_f V_f = n R T_f = P_i V_i = P_ext V_f\_ext\). As \(P_i = P_f = P_ext\), we know that \(V_i = V_f\_ext\). For the work done in this case, we just have to apply the work done agenda by the gas formula \(W = -P_ext * ΔV = -P_ext * (V_f\_ext - V_i)\).

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