The internal energy of an ideal gas decreases by the same amount as the work done by the system.

1. Thermodynamics:

Study of energy changes between system & surrounding is called thermodynamics. There are three types of systems open, closed & isolated.

2. Zeroth Law of Thermodynamics:

If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other.

3. First Law of Thermodynamics:

Heat supplied $\left(Q\right)$ to a system is equal to algebraic sum of change in internal energy $\left(\mathrm{\Delta }U\right)$ of the system and mechanical work $\left(W\right)$ done by the system.

(i) $Q=W+\mathrm{\Delta }U$ $\left[W=\int PdV ; \mathrm{\Delta }U=n{C}_v\mathrm{\Delta }T\right]$

(ii) Sign Convention:

(a) Heat absorbed by the system → positive.

(b) Heat rejected by the system → negative.

(c) Increase in internal energy (i.e. rise in temperature) → positive.

(d) Decrease in internal energy (i.e. fall in temperature) → negative.

(e) Work done by the system → positive.

(f) Work done on the system → negative.

(iii) For isochoric process:

(a) In this process, V=constant

(b) P∝T

(c) W=0,

(d) $Q=\Delta U=\mu C_v\mathrm{\Delta }T$

(iv) For isobaric process:

(a) In this process, P=constant

(b) $V\propto T$

(c) $Q=\mu C_p\mathrm{\Delta }T$

(d) $\mathrm{\Delta }U=\mu C_v\mathrm{\Delta }T$

(e) $W=P\left(V_2-V_1\right)=\mu R\mathrm{\Delta }T$

(v) For adiabatic process:

(a) In this process, Q=0

(b) $P{V}^{\gamma }=$ constant

(c) $T^{\gamma }{P}^{1-\gamma }=$ constant

(d) $T{V}^{\gamma -1}=$ constant

(e) $W=-\mathrm{\Delta }U=\mu C_V\left(T_1-T_2\right)=\frac{P_1{V}_1-P_2{V}_2}{\gamma -1}$

(vi) For isothermal process:

(a) In this process, T= constant or $\mathrm{\Delta }T=0$

(b) PV= constant

(c) $\mathrm{\Delta }U=\mu C_v\mathrm{\Delta }T=0$

(d) $Q=W=\mu RT\ln \left(\frac{V_2}{V_1}\right)=\mu RT\ln \left(\frac{P_1}{P_2}\right)$

(vii) For any general polytropic process:

(a) PVx= constant

(b) Molar heat capacity, C=Cv+R1-x

(c) Work done by gas, $W=\frac{nR\left(T_1-T_2\right)}{x-1}=\frac{\left(P_1{V}_1-P_2{V}_2\right)}{x-1}$

(d) Slope of $P-V$ diagram (also known as indicator diagram) at any point is $\frac{\mathrm{d}P}{\mathrm{d}V}=-x\frac{P}{V}$

(viii) For cyclic process,

(a) $\mathrm{\Delta }U=0$

(b) Q=W

4. Second Law of Thermodynamics:

(i) Kelvin- Planck Statement: 

It is impossible to construct an engine, operating in a cycle, which will produce no effect other than extracting heat from a reservoir and performing an equivalent amount of work.

(ii) Rudlope Classius Statement:

It is impossible to make heat flow from a body at a lower temperature to a body at a higher temperature without doing external work on the working substance.

5. Heat Engines and Refrigerators:

(i) Carnot Engine: It is a hypothetical engine with maximum possible efficiency.

Process $1\to 2 \text{\&} 3\to 4$ are isothermal.

Process $2\to 3 \text{\&} 4\to 1$ are adiabatic.

(ii) Efficiency of a cycle:

(a) For Carnot cycle:

$\eta =\frac{\mathrm{W}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{ }\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{ }\mathrm{b}\mathrm{y}\mathrm{ }\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{ }\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}}{\mathrm{H}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{ }\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{d}}\mathrm{ }=\frac{W}{Q_1}=\frac{Q_1-Q_2}{Q_1}=1-\frac{Q_2}{Q_1}=1-\frac{T_2}{T_1}$

(b) For refrigerator:

Coefficient of performance, $\beta =\frac{Q_2}{W}=\frac{Q_2}{Q_1-Q_2}=\frac{T_2}{T_1-T_2}$

6. Bulk modulus of gases:

(i) $B=\frac{\mathrm{\Delta }P}{-\frac{\mathrm{\Delta }V}{V}}$

(ii) Isothermal bulk modulus of elasticity, $B_{\mathrm{Iso}}=-V{\left(\frac{\partial P}{\partial V}\right)}_{\mathrm{T}=\mathrm{constant}}$

(iii) Adiabatic bulk modulus of elasticity, $B_{\mathrm{Adia}}=-\gamma V\left(\frac{\partial P}{\partial V}\right)\Rightarrow B_{\mathrm{Adia}}=\gamma B_{\mathrm{Iso}}$

7. Thermodynamic Processes and Indicator Diagrams:

(i) Work done is least for monoatomic gas (adiabatic process)

(ii) Air quickly leaking out of a balloon becomes cooler as the leaking air undergoes adiabatic expansion.

(iii) First law of thermodynamics does not forbid flow of heat from lower temperature to higher temperature.

(iv) First law of thermodynamics allows many processes which actually don't happen.