Question

Figure 8.8 shows the $P$ versus $V$ graphs of three expansions: (a) the reversible isothermal, (b) the irreversible isothermal, and (c) the reversible adiabatic. Plot the corresponding curves for $T$ versus $S$ of these three processes, letting all three processes start from the same initial point ( $T_1, S_1$ ) in the middle of your graph. Be quantitative if possible; otherwise, sketch an approximate curve. Label the curves $\mathrm{a}, \mathrm{b}$, and c . THINKING AHEAD $>[$ How should the areas under these curves compare?] 

   Figure 8.8 shows the $P$ versus $V$ graphs of three expansions: (a) the reversible isothermal, (b) the irreversible isothermal, and (c) the reversible adiabatic. Plot the corresponding curves for $T$ versus $S$ of these three processes, letting all three processes start from the same initial point ( $T_1, S_1$ ) in the middle of your graph. Be quantitative if possible; otherwise, sketch an approximate curve. Label the curves $\mathrm{a}, \mathrm{b}$, and c . THINKING AHEAD $>[$ How should the areas under these curves compare?]
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Physical Chemistry : Thermodynamics, Statistical Mechanics & Kinetics
Physical Chemistry : Thermodynamics, Statistical Mechanics & Kinetics
Andrew Cooksy 1st Edition
Chapter 9, Problem 24 ↓
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Figure 8.8 shows the $P$ versus $V$ graphs of three expansions: (a) the reversible isothermal, (b) the irreversible isothermal, and (c) the reversible adiabatic. Plot the corresponding curves for $T$ versus $S$ of these three processes, letting all three processes start from the same initial point ( $T_1, S_1$ ) in the middle of your graph. Be quantitative if possible; otherwise, sketch an approximate curve. Label the curves $\mathrm{a}, \mathrm{b}$, and c . THINKING AHEAD $>[$ How should the areas under these curves compare?]
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(a) From ideal gas law $p=\left(\frac{v R}{V}\right) T=k T\left(\right.$ where $\left.k=\frac{v R}{V}\right)$ For isochoric process, obviously $k=$ constant, thus $p=k T$, represents a straight line passing through the origin and its slope becomes $k$. For isobaric process $p=$ constant, thus on $p-T$ curve, it is a horizontal straight line parallel to $T$ -axis, if $T$ is along horizontal (or $x$ -axis) For isothermal process, $T=$ constant, thus on $p-T$ curve, it represents a vertical straight line if $T$ is taken along horizontal (or $x$ -axis) For adiabatic process $T^{\gamma} p^{1-\gamma}=$ constant After diffrentiating, we get $(1-\gamma) p^{-\gamma} d p \cdot T^{\gamma}+\gamma p^{1-\gamma} \cdot T^{\gamma-1} \cdot d T=0$ $$ \frac{d p}{d T}=\left(\frac{\gamma}{1-\gamma}\right)\left(\frac{p^{1-\gamma}}{p^{-\gamma}}\right)\left(\frac{T^{\gamma-1}}{T^{\psi}}\right)=\left(\frac{\gamma}{\gamma-1}\right) \frac{p}{T} $$ The approximate plots of isochoric, isobaric, isothermal, and adiabatic processess are drawn in the answersheet. (b) As $p$ is not considered as variable, we have from ideal gas law $$ V=\frac{v R}{p} T=k^{\prime} T\left(\text { where } k^{\prime}=\frac{v R}{p}\right) $$ On $V-T$ co-ordinate system let us, take $T$ along $x$ -axis. For isochoric process $V=$ constant, thus $k^{\prime}=$ constant and $V=k^{\prime} T$ obviously represents a straight line pasing through the origin of the co-ordinate system and $k^{\prime}$ is its slope. For isothermal process $T=$ constant. Thus on the stated co- ordinate system it represents a straight line parallel to the $V$ - axis. For adiabatic process $T V^{\gamma-1}=$ constant After differentiating, we get $(\gamma-1) V^{\gamma-2} d V \cdot T+V^{\dagger-1} d T=0$ $$ \frac{d V}{d T}=-\left(\frac{1}{\gamma-1}\right) \cdot \frac{V}{T} $$ The approximate plots of isochoric, isobaric, isothermal and adiabatic processess are drawn in the answer sheet.

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Figure $P 16.54$ shows the $P-V$ diagrams for a variety of different processes involving an ideal gas. a. Which diagrams describe an expansion? b. Which diagrams describe a compression? c. Which diagrams might describe an adiabatic process? d. Which diagrams might describe an isothermal process?

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Transcript

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00:04 Hi everyone for ideal guess.
00:11 Gas equation is p is called to new r t by v.
00:19 Here we can write it is k into t.
00:27 K having the value new r upon volume for isochoric process k is constant.
00:46 P is called to k t so it will be a straight line graph for isobaric process, pressure is constant.
01:14 Hence, pt graph is parallel to temperature axis...
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