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Chapter 20: Problem 4
An ideal gas undergoes an isothermal expansion. What will happen to its entropy? a) It will increase. c) It's impossible to determine. b) It will decrease. d) It will remain unchanged.
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While looking at a very small system, a scientist observes that the entropy of the system spontaneously decreases. If true, is this a Nobel-winning discovery or is it not that significant?
An ideal gas is enclosed in a cylinder with a movable piston at the top. The walls of the cylinder are insulated, so no heat can enter or exit. The gas initially occupies volume \(V_{1}\) and has pressure \(p_{1}\) and temperature \(T_{1}\). The piston is then moved very rapidly to a volume of \(V_{2}=3 V_{1}\). The process happens so rapidly that the enclosed gas does not do any work. Find \(p_{2}, T_{2},\) and the change in entropy of the gas.
Consider a room air conditioner using a Carnot cycle at maximum theoretical efficiency and operating between the temperatures of \(18^{\circ} \mathrm{C}\) (indoors) and \(35^{\circ} \mathrm{C}\) (outdoors). For each 1.00 J of heat flowing out of the room into the air conditioner: a) How much heat flows out of the air conditioner to the outdoors? b) By approximately how much does the entropy of the room decrease? c) By approximately how much does the entropy of the outdoor air increase?
A key feature of thermodynamics is the fact that the internal energy, \(E_{\text {int }}\) of a system and its entropy, \(S\), are state variables; that is, they depend only on the thermodynamic state of the system and not on the processes by which it reached that state (unlike, for example, the heat content, \(Q\) ). This means that the differentials \(d E_{\text {int }}=T d S-p d V\) and \(d S=\) \(T^{-1} d E_{\text {int }}+p T^{-1} d V,\) where \(T\) is temperature (in kelvins), \(p\) is pressure, and \(V\) is volume, are exact differentials as defined in calculus. What relationships follow from this fact?
The entropy of a macroscopic state is given by \(S=k_{B} \ln w\) where \(k_{\mathrm{B}}\) is the Boltzmann constant and \(w\) is the number of possible microscopic states. Calculate the change in entropy when \(n\) moles of an ideal gas undergo free expansion to fill the entire volume of a box after a barrier between the two halves of the box is removed.
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