Total Translational K.E. Of Gas

$$=\frac{1}{2} M\left\langle v^{2}\right\rangle=\frac{3}{2} P V=\frac{3}{2} n R T$$

$$<v^{2}>=\frac{3 P}{\rho} \quad v_{rms}=\sqrt{\frac{3 P}{\rho}}=\sqrt{\frac{3 RT}{M_{mol}}}=\sqrt{\frac{3 KT}{m}}$$

Important Points :

$$-\mathrm{v}_{\mathrm{rms}} \propto \sqrt{\mathrm{T}}$$

$$\bar{v}=\sqrt{\frac{8 K T}{\pi m}}=1.59 \sqrt{\frac{K T}{m}}$$

$$\mathrm{v}_{\mathrm{rms}}=1.73 \sqrt{\frac{\mathrm{KT}}{\mathrm{m}}}$$

Most probable speed:

$$V_{p}=\sqrt{\frac{2 KT}{m}}=1.41 \sqrt{\frac{KT}{m}} \therefore V_{rms}>\overline{V}>V_{mp}$$

Degree of freedom :

• Monoatomic $f=3$

• Diatomic $f=5$

• Polyatomic $f=6$

Maxwell’s Law Of Equipartition Of Energy :

• Total K.E. of the molecule $=\frac{1}{2} \mathrm{fKT}$

• For an ideal gas : Internal energy $$U=\frac{f}{2} n R T$$

Ideal gas law:

$$PV = nRT$$

First Law Of Thermodynamics:

$$\Delta U = Q - W$$

Isothermal Process:

• Workdone in isothermal process : $$W=\left[2.303 nRT \log _{10} \frac{V_f}{V_i}\right]$$

• Internal energy in isothermal process : $$ \Delta \mathrm{U}=0$$

Isochoric Process

• Work done in isochoric process : $$ d W=0$$

• Change in internal energy for an ideal gas during an isochoric process: $$\Delta U = nC_v\Delta T$$

Isobaric process :

• Work done: $$\Delta \mathrm{W}=n R\left(T_{\mathrm{f}}-\mathrm{T}_{\mathrm{i}}\right)$$

• Change in internal energy for isobaric process: $$\Delta U = n C_p \Delta T$$

Adiabatic process :

• Work done: $$\Delta W=\frac{nR\left(T_{i}-T_{f}\right)}{\gamma-1}$$

Specific heat :

$$ C_{V}=\frac{f}{2} R \quad C p=\left(\frac{f}{2}+1\right) R $$

Molar heat capacity of ideal gas in terms of $\mathbf{R}$ :

(i) for monoatomic gas : $$\frac{C_{p}}{C_{v}}=1.67$$

(ii) for diatomic gas : $$\frac{C_{p}}{C_{v}}=1.4$$

(iii) for triatomic gas : $$\frac{C_{p}}{C_{v}}=1.33$$

Heat Capacity Ratio :

$$\gamma=\frac{C_{p}}{C_{v}}=\left[1+\frac{2}{f}\right]$$

Mayer’s equation:

For ideal gas: $$C_{p}-C_{v}=R $$

In cyclic process :

$$\Delta Q=\Delta W$$

In a mixture of non-reacting gases :

$$\text{Molar weight}=\frac{n_{1} M_{1}+n_{2} M_{2}}{n_{1}+n_{2}}$$

$$C_{v}=\frac{n_{1} C_{v_{1}}+n_{2} C_{v_{2}}}{n_{1}+n_{2}}$$

$$\gamma=\frac{C_{p(\text { mix })}}{C_{v(\text { mix })}}=\frac{n_{1} C_{p_{1}}+n_{2} C_{p_{2}}+\ldots}{n_{1} C_{v_{1}}+n_{2} C_{v_{2}}+\ldots}$$

Heat Engines

$$\text{Efficiency,} \eta=\frac{\text { work done by the engine }}{\text { heat sup plied to it }}$$

$$\eta=\frac{W}{Q_{H}}=\frac{Q_{H}-Q_{L}}{Q_{H}}=1-\frac{Q_{L}}{Q_{H}}$$

Second law of Thermodynamics

• 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.

• Clausius 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

Entropy:

• Change in entropy of the system is $$\Delta S=\frac{\Delta Q}{T} \Rightarrow S_{f}-S_{i}=\int_{i}^{f} \frac{\Delta Q}{T}$$
• In an adiabatic reversible process, entropy of the system remains constant.

Efficiency of Carnot Engine:

(1) Operation I (Isothermal Expansion)

(2) Operation II (Adiabatic Expansion)

(3) Operation III (Isothermal Compression)

(4) Operation IV (Adiabatic Compression)

Thermal Efficiency of a Carnot Engine:

$$\frac{V_{2}}{V_{1}}=\frac{V_{3}}{V_{4}} \Rightarrow \frac{Q_{2}}{Q_{1}}=\frac{T_{2}}{T_{1}} \Rightarrow \eta=1-\frac{T_{2}}{T_{1}}$$

Refrigerator (Heat Pump)

• Coefficient of performance, $$\beta=\frac{Q_{2}}{W}=\frac{1}{\frac{T_{1}}{T_{2}}-1}==\frac{1}{\frac{T_{1}}{T_{2}}-1}$$

Calorimetry And Thermal Expansion Types Of Thermometers :

(a) Liquid Thermometer : $$ T=\left[\frac{\ell-\ell_{0}}{\ell_{100}-\ell_{0}}\right] \times 100$$

(b) Gas Thermometer :

• Constant volume : $$T=\left[\frac{P-P_{0}}{P_{100}-P_{0}}\right] \times 100 ; P=P_{0}+\rho g h$$

• Constant Pressure : $$T=\left[\frac{\mathrm{V}}{\mathrm{V}-\mathrm{V}^{\prime}}\right] \mathrm{T}_{0}$$

(c) Electrical Resistance Thermometer :

$$ T=\left[\frac{R_{t}-R_{0}}{R_{100}-R_{0}}\right] \times 100 $$

Thermal Expansion :

(a) Linear :

$$ \alpha=\frac{\Delta L}{L_{0} \Delta T} \quad \text { or } \quad L=L_{0}(1+\alpha \Delta T) $$

(b) Area/superficial :

$$ \beta=\frac{\Delta A}{A_{0} \Delta T} \quad \text { or } \quad A=A_{0}(1+\beta \Delta T) $$

(c) volume/ cubical :

$$ r=\frac{\Delta V}{V_{0} \Delta T} \quad \text { or } \quad V=V_{0}(1+\gamma \Delta T) $$

$$ \alpha=\frac{\beta}{2}=\frac{\gamma}{3} $$

Thermal stress of a material :

$$ \frac{F}{A}=Y \frac{\Delta \ell}{\ell} $$

Energy stored per unit volume :

$$ E=\frac{1}{2} K(\Delta L)^{2} \quad \text { or } \quad E=\frac{1}{2} \frac{A Y}{L}(\Delta L)^{2} $$

Variation of time period of pendulum clocks :

$$ \Delta \mathrm{T}=\frac{1}{2} \alpha \Delta \theta \mathrm{T} $$

$$ \mathrm{T}^{\prime}<\mathrm{T} \quad \text { - clock-fast : time-gain } $$

$$ \mathrm{T}^{\prime}>\mathrm{T} \quad \text { - clock slow : time-loss } $$

Calorimetry :

• Specific heat: $$S=\frac{\mathrm{Q}}{\mathrm{m} \cdot \Delta \mathrm{T}}$$

• Molar specific heat: $$\mathrm{C}=\frac{\Delta \mathrm{Q}}{\mathrm{n} \cdot \Delta \mathrm{T}}$$

• Water equivalent: $$m_{m} S_{m}=m_{w} S_{w}$$

Heat Transfer

• Thermal Conduction : $$ \frac{d Q}{d t}=-K A \frac{d T}{d x}$$

• Thermal Resistance : $$\mathrm{R}=\frac{\ell}{\mathrm{KA}}$$

Series And Parallel Combination Of Rod :

(i) Series : $$\frac{\ell_{\text {eq }}}{K_{\text {eq }}}=\frac{\ell_{1}}{K_{1}}+\frac{\ell_{2}}{K_{2}}+\ldots$$ when $$\left(A_{1}=A_{2}=A_{3}=\ldots\right)$$

(ii) Parallel : $$K_{\text {eq }} A_{e q}=K_{1} A_{1}+K_{2} A_{2}+\ldots$$ when $$\left(\ell_{1}=\ell_{2}=\ell_{3}=\ldots\right)$$

For absorption, reflection and transmission: $ r+t+a=1 $

• Emissive power : $$ \mathrm{E}=\frac{\Delta \mathrm{U}}{\Delta \mathrm{A} \Delta \mathrm{t}}$$

• Spectral emissive power : $$ E_{\lambda}=\frac{d E}{d \lambda}$$

Emissivity : $$ e=\frac{E \text { of a body at } \mathrm{T} \text { temperature }}{\mathrm{E} \text { of a black body at } \mathrm{T} \text {temperature}}$$

• Kirchoff’s law : $$4\frac{E \text { (body) }}{a \text { (body) }}=E \text{(black body)}$$

Wein’s Displacement law :

$$\lambda_{\mathrm{m}} \cdot \mathrm{T}=\mathrm{b}$$

$$b=0.282 \mathrm{~cm}-\mathrm{k}$$

Stefan Boltzmann law :

$$ \mathrm{u}=\sigma \mathrm{T}^{4} \quad \quad \mathrm{~s}=5.67 \times 10^{-8} \mathrm{W} \mathrm{m}^{2} \mathrm{k}^{4} $$

$$ \Delta u=u-u_{0}=e \sigma A \left(T^{4}-T_{0}^{4}\right) $$

Newton’s law of cooling :

$$\frac{\mathrm{d} \theta}{\mathrm{dt}}=\mathrm{k}\left(\theta-\theta_{0}\right) ; \quad \theta=\theta_{0}+\left(\theta_{\mathrm{i}}-\theta_{0}\right) \mathrm{e}^{-\mathrm{kt}}$$