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Chapter 15: Q10P (page 412)

Question: (II) Consider the following two-step process. Heat is allowed to flow out of an ideal gas at constant volume so that its pressure drops from 2.2 atm to 1.4 atm. Then the gas expands at constant pressure, from a volume of 5.9 L to 9.3 L, where the temperature reaches its original value. See Fig.15鈥�22. Calculate (a) the total work done by the gas in the process, (b) the change in internal energy of the gas in the process, and (c) the total heat flow into or out of the gas.

Short Answer

Expert verified

(a) The total work done in two-step process is \(480\;{\rm{J}}\).

(b) The change in internal energy of the gas in the process is \(0\).

(c) The heat absorbed in this two-step process is \(480\;{\rm{J}}\).

Step by step solution

01

Understanding of constant volume process

In the constant volume process, the reaction's heat is equivalent to the variation in the internal energy since the value of the work done in this process is equal to zero.

02

Evaluation of the total work done by the gas in the two step process

(a)

In the first step from a to be, the heat is allowed to flow out of an ideal gas at constant volume. So, the change in the volume in this process will be \(\Delta {V_{{\rm{ab}}}} = 0\). Thus, the value of the work done in this process will be:

\(\begin{aligned}{c}{W_{{\rm{ab}}}} &= P\Delta {V_{{\rm{ab}}}}\\{W_{{\rm{ab}}}} &= 0\end{aligned}\)

The process b to c is a constant pressure process. In this process, the pressure is:

\(\begin{aligned}{c}P &= \left( {1.4\;{\rm{atm}}} \right)\left( {\frac{{1.01 \times {{10}^5}\;{\rm{Pa}}}}{{1\;{\rm{atm}}}}} \right)\\P &= 1.414 \times {10^5}\;{\rm{Pa}}\end{aligned}\)

The change in the volume is:

\(\begin{aligned}{c}\Delta {V_{{\rm{bc}}}} &= \left( {\left( {9.3\;{\rm{L}}} \right) - \left( {5.9\;{\rm{L}}} \right)} \right)\left( {\frac{{{{10}^{ - 3}}\;{{\rm{m}}^{\rm{3}}}}}{{1\;{\rm{L}}}}} \right)\\\Delta {V_{{\rm{bc}}}} &= 3.4 \times {10^{ - 3}}\;{{\rm{m}}^{\rm{3}}}\end{aligned}\)

So, the work done in process b to c will be:

\(\begin{aligned}{c}{W_{{\rm{bc}}}} &= P\Delta {V_{{\rm{bc}}}}\\{W_{{\rm{bc}}}} &= \left( {1.414 \times {{10}^5}\;{\rm{Pa}}} \right)\left( {3.4 \times {{10}^{ - 3}}\;{{\rm{m}}^{\rm{3}}}} \right)\\{W_{{\rm{bc}}}} &= 480\;{\rm{J}}\end{aligned}\)

The total work done in two-step process can be calculated as:

\(\begin{aligned}{c}W &= {W_{{\rm{bc}}}} + {W_{{\rm{ab}}}}\\W &= \left( {480\;{\rm{J}}} \right) + 0\\W &= 480\;{\rm{J}}\end{aligned}\)

Thus, the total work done in two-step process is \(480\;{\rm{J}}\).

03

Evaluation of the internal energy of the gas in this process

(b)

The expression for the internal energy of a monoatomic gas is as follows:

\(\Delta U = \frac{3}{2}nR\Delta T\)

In this two-step process, there is no change in the temperature. Therefore, the value of the change in the internal energy will be:

\(\Delta U = 0\)

Thus, the change in internal energy of the gas in the process is \(0\).

04

Evaluation of the total heat that flow into or out of the gas

(c)

Thetotal heat flow into or out of the gas can be calculated as:

\(\begin{aligned}{c}\Delta U &= Q - W\\0 &= Q - \left( {480\;{\rm{J}}} \right)\\Q &= 480\;{\rm{J}}\end{aligned}\)

Since, the sign is positive, the heat flows into the system.

Thus, the heat absorbed in this two-step process is \(480\;{\rm{J}}\).

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Most popular questions from this chapter

Question: An ideal gas undergoes an isobaric compression and then an isovolumetric process that brings it back to its initial temperature. Had the gas undergone one isothermal process instead,

(a) the work done on the gas would be the same.

(b) the work done on the gas would be less.

(c) the work done on the gas would be greater.

(d) Need to know the temperature of the isothermal process.

Question: An ideal gas undergoes an isothermal expansion from state A to state B. In this process (use sign conventions, page 413),

(a) \[Q = 0,\;\Delta U = 0,\;W > 0\].

(b) \[Q > 0,\;\Delta U = 0,\;W < 0\].

(c) \[Q = 0,\;\Delta U > 0,\;W > 0\].

(d) \[Q > 0,\;\Delta U = 0,W > 0\].

(e) \[Q = 0,\;\Delta U < 0,\;W < 0\].

A 38% efficient power plant puts out 850 MW of electrical power. Cooling towers take away the exhaust heat.

(a) If the air \(\left( {{\bf{k}}{{\bf{m}}{\bf{3}}}} \right)\) temperature is allowed to rise by 7.0掳C, estimate what volume of air is heated per day. Will the local climate be heated significantly?

(b) If the heated air were to form a layer 180 m thick, estimate how large an area it would cover for 24 h of operation. Assume the air has a density of \({\bf{1}}{\bf{.3 kg/}}{{\bf{m}}{\bf{3}}}\) and a specific heat of about \({\bf{1}}{\bf{.0}}{\rm{ }}{\bf{kJ/kg}} \cdot {\bf{\circ C}}\) at constant pressure.

Question: (II) An ideal gas expands at a constant total pressure of 3.0 atm from 410 mL to 690 mL. Heat then flows out of the gas at constant volume, and the pressure and temperature are allowed to drop until the temperature reaches its original value. Calculate (a) the total work done by the gas in the process, and (b) the total heat flow into the gas.

Question:(II) A four-cylinder gasoline engine has an efficiency of 0.22 and delivers 180 J of work per cycle per cylinder. If the engine runs at 25 cycles per second (1500 rpm), determine (a) the work done per second, and (b) the total heat input per second from the gasoline. (c) If the energy content of gasoline is 130 MJ per gallon, how long does one gallon last?
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