Heat And Thermodynamics
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 :
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Monoatomic $f=3$
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Diatomic $f=5$
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Polyatomic $f=6$
Maxwell’s Law Of Equipartition Of Energy :
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Total K.E. of the molecule $=\frac{1}{2} \mathrm{fKT}$
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For an ideal gas : Internal energy $$U=\frac{f}{2} n R T$$
Ideal gas law:
PYQ-2023-Thermodynamics-Q9, PYQ-2023-Kinetic-Theory-Of-Gases-Q2
$$PV = nRT$$
First Law Of Thermodynamics:
PYQ-2023-Thermodynamics-Q10, PYQ-2023-Thermodynamics-Q13, PYQ-2023-Thermodynamics-Q16
$$\Delta U = Q - W$$
Isothermal Process:
PYQ-2023-Thermodynamics-Q3, PYQ-2023-Thermodynamics-Q6
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Workdone in isothermal process : $$W=\left[2.303 nRT \log _{10} \frac{V_f}{V_i}\right]$$
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Internal energy in isothermal process : $$ \Delta \mathrm{U}=0$$
Isochoric Process
PYQ-2023-Thermodynamics-Q6, PYQ-2023-Thermodynamics-Q20
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Work done in isochoric process : $$ d W=0$$
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Change in internal energy for an ideal gas during an isochoric process: $$\Delta U = nC_v\Delta T$$
Isobaric process :
PYQ-2023-Thermodynamics-Q1, PYQ-2023-Thermodynamics-Q6, PYQ-2023-Thermodynamics-Q18
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Work done: $$\Delta \mathrm{W}=n R\left(T_{\mathrm{f}}-\mathrm{T}_{\mathrm{i}}\right)$$
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Change in internal energy for isobaric process: $$\Delta U = n C_p \Delta T$$
Adiabatic process :
PYQ-2023-Thermodynamics-Q6, PYQ-2023-Thermodynamics-Q17, PYQ-2023-Thermodynamics-Q19
- Work done: $$\Delta W=\frac{nR\left(T_{i}-T_{f}\right)}{\gamma-1}$$
Specific heat :
PYQ-2023-Thermodynamics-Q12, PYQ-2023-Thermal-Properties-Of-Matter-Q3
$$ C_{V}=\frac{f}{2} R \quad C p=\left(\frac{f}{2}+1\right) R $$
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 :
PYQ-2023-Thermodynamics-Q2, PYQ-2023-Thermodynamics-Q15
$$\gamma=\frac{C_{p}}{C_{v}}=\left[1+\frac{2}{f}\right]$$
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:
PYQ-2023-Thermodynamics-Q4, PYQ-2023-Thermodynamics-Q14, PYQ-2023-Thermodynamics-Q21
(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 :
PYQ-2023-Thermodynamics-Q8, PYQ-2023-Thermal-Properties-Of-Matter-Q2, PYQ-2023-Kinetic-Theory-Of-Gases-Q4
(a) Liquid Thermometer : $$ T=\left[\frac{\ell-\ell_{0}}{\ell_{100}-\ell_{0}}\right] \times 100$$
(b) Gas Thermometer :
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Constant volume : $$T=\left[\frac{P-P_{0}}{P_{100}-P_{0}}\right] \times 100 ; P=P_{0}+\rho g h$$
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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 :
PYQ-2023-Thermal-Properties-Of-Matter-Q1
$$ \alpha=\frac{\Delta L}{L_{0} \Delta T} \quad \text { or } \quad L=L_{0}(1+\alpha \Delta T) $$
(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 :
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Specific heat: $$S=\frac{\mathrm{Q}}{\mathrm{m} \cdot \Delta \mathrm{T}}$$
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Molar specific heat: $$\mathrm{C}=\frac{\Delta \mathrm{Q}}{\mathrm{n} \cdot \Delta \mathrm{T}}$$
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Water equivalent: $$m_{m} S_{m}=m_{w} S_{w}$$
Heat Transfer
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Thermal Conduction : $$ \frac{d Q}{d t}=-K A \frac{d T}{d x}$$
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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 $
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Emissive power : $$ \mathrm{E}=\frac{\Delta \mathrm{U}}{\Delta \mathrm{A} \Delta \mathrm{t}}$$
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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 :
$$\frac{\mathrm{d} \theta}{\mathrm{dt}}=\mathrm{k}\left(\theta-\theta_{0}\right) ; \quad \theta=\theta_{0}+\left(\theta_{\mathrm{i}}-\theta_{0}\right) \mathrm{e}^{-\mathrm{kt}}$$