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Chapter 18: Problem 52

Prove that the slope of an adiabat at a given point in a \(p V\) diagram is \(\gamma\) times the slope of the isotherm passing through the same point.

Short Answer

Expert verified
The slope of an adiabat at a given point in a \(pV\) diagram is indeed \(\gamma\) times the slope of the isotherm passing through the same point. This conclusion is drawn from understanding the mathematical expressions for adiabatic and isothermal processes and comparing their slopes.

Step by step solution

01

Define the adiabatic process

In an adiabatic process, the ideal gas law transforms to \(pV^\gamma = \text{constant}\), where \(p\) is pressure, \(V\) is volume and \(\gamma\) is the ratio of specific heats, given as \(C_p / C_v\). Taking the total differential, we get \((\gamma pV^{\gamma - 1})dV + V^\gamma dp = 0\). We can solve for \(dp/ dV\) to get the slope of the adiabat on the \(pV\) diagram: \(\frac{dp}{dV} = -\gamma \frac{p}{V}\).
02

Define the isothermal process

In an isothermal process, the ideal gas law becomes \(pV = \text{constant}\). Again, taking the total differential, we obtain \(V dp + p dV = 0\). Solving for \(dp/ dV\), we find the slope of the isotherm: \( \frac{dp}{dV} = - \frac{p}{V}\).
03

Show the relationship

By comparing the slopes obtained from the adiabatic and isothermal processes, we can see that the slope of an adiabat at a given point in a \(pV\) diagram is \(\gamma\) times higher than the slope of the isotherm passing through the same point. This is because the slope of the adiabat is \(-\gamma p / V\) while the slope of the isotherm is \(-p / V\). Thus, the slope of the adiabat is \(\gamma\) times the slope of the isotherm.

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Most popular questions from this chapter

A \(3.50-\mathrm{mol}\) sample of ideal gas with molar specific heat \(C_{V}=\frac{5}{2} R\) is initially at \(255 \mathrm{~K}\) and \(101 \mathrm{kPa}\) pressure. Determine the final temperature and the work done by the gas when \(1.75 \mathrm{~kJ}\) of heat are added to the gas (a) isothermally, (b) at constant volume, and (c) isobarically.

By what factor must the volume of a gas with \(\gamma=1.4\) be changed in an adiabatic process if the pressure is to double?

A power plant extracts thermal energy from its fuel at the rate of \(3810 \mathrm{MW}\) and produces electrical energy at the rate of \(1250 \mathrm{MW}\). There's a proposal to use the waste heat from this plant to heat nearby homes. If the average home requires \(43.2 \mathrm{GJ}\) of energy in a winter month, how many homes could be served if \(100 \%\) of the waste heat from the power plant were available for home heating?

A research balloon is prepared for launch by pumping into it \(1.75 \times 10^{3} \mathrm{~m}^{3}\) of helium gas at \(12^{\circ} \mathrm{C}\) and \(1.00\) atm pressure. It rises high into the atmosphere to where the pressure is only \(0.340 \mathrm{~atm}\). Assuming the balloon doesn't exchange significant heat with its surroundings, find (a) its volume and (b) its temperature at the higher altitude.

It takes 0.8 kJ to compress a gas isothermally to half its original volume. How much work would it take to compress it by a factor of 15 starting from its original volume?
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