Thermodynamic Processes — Revision Notes

⚡ 30-Second Revision

First Law: — $Delta U = Q - W$
Internal Energy (Ideal Gas): — $Delta U = nC_vDelta T$
Work Done: — $W = int P dV$
Isobaric ($P= ext{const}$): — $W = P(V_2-V_1)$, $Q = nC_pDelta T$
Isochoric ($V= ext{const}$): — $W = 0$, $Q = nC_vDelta T$
Isothermal ($T= ext{const}$): — $Delta U = 0$, $W = nRT ln(V_2/V_1)$, $Q = W$
Adiabatic ($Q=0$): — $PV^gamma = \text{const}$, $TV^{gamma-1} = \text{const}$, $P^{1-gamma}T^gamma = \text{const}$, $W = \frac{P_1V_1 - P_2V_2}{gamma-1} = nC_v(T_1-T_2)$
Cyclic Process: — $Delta U_{cycle} = 0$, $Q_{net} = W_{net}$
Adiabatic Index: — $gamma = C_p/C_v$
Mayer's Relation: — $C_p - C_v = R$

2-Minute Revision

Thermodynamic processes describe how a system changes state, involving heat ($Q$) and work ($W$) transfers. The First Law, $Delta U = Q - W$, is central. For an ideal gas, internal energy ($Delta U$) depends only on temperature change ($Delta T$), specifically $Delta U = nC_vDelta T$.

Isobaric (Constant Pressure): — $P$ is constant. Work $W = PDelta V$. Heat $Q = nC_pDelta T$. P-V diagram is a horizontal line.
Isochoric (Constant Volume): — $V$ is constant. Work $W = 0$. Heat $Q = nC_vDelta T$. P-V diagram is a vertical line.
Isothermal (Constant Temperature): — $T$ is constant. For ideal gas, $Delta U = 0$. Work $W = nRT ln(V_2/V_1)$. Heat $Q = W$. P-V diagram is a hyperbola.
Adiabatic (No Heat Exchange): — $Q = 0$. Work $W = -Delta U$. $PV^gamma = \text{constant}$. P-V diagram is steeper than isothermal.
Cyclic Process: — System returns to initial state. $Delta U_{cycle} = 0$, so $Q_{net} = W_{net}$. Work is the area enclosed on a P-V diagram. Remember sign conventions: $Q$ positive if added, $W$ positive if done by system.

5-Minute Revision

Thermodynamic processes are pathways for a system to transition between states, defined by changes in state variables like pressure ($P$), volume ($V$), and temperature ($T$). Energy is exchanged as heat ($Q$) and work ($W$). The First Law of Thermodynamics, $Delta U = Q - W$, is the guiding principle, where $Delta U$ is the change in internal energy. For an ideal gas, $Delta U$ is solely dependent on temperature change, given by $Delta U = nC_vDelta T$.

Let's quickly review the key processes:

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Isobaric Process (Constant Pressure):

* $P = \text{constant}$. * Work done: $W = P(V_2 - V_1)$. This is simply pressure times the change in volume. * Heat exchanged: $Q = nC_pDelta T$. Here, $C_p$ is molar specific heat at constant pressure. * P-V diagram: A horizontal line.

1
Isochoric Process (Constant Volume):

* $V = \text{constant}$. * Work done: $W = 0$, because there's no change in volume. * Heat exchanged: $Q = nC_vDelta T$. All heat goes into changing internal energy. * P-V diagram: A vertical line.

1
Isothermal Process (Constant Temperature):

* $T = \text{constant}$. * For an ideal gas, $Delta U = 0$ since $U$ depends only on $T$. * Work done: $W = nRT ln(V_2/V_1)$. Also $W = P_1V_1 ln(V_2/V_1)$. * Heat exchanged: $Q = W$ (from $Delta U = Q - W = 0$). * P-V diagram: A hyperbola ($PV = \text{constant}$). Example: A gas expands slowly in contact with a heat reservoir.

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Adiabatic Process (No Heat Exchange):

* $Q = 0$. * From First Law: $Delta U = -W$. Work done by the gas comes from its internal energy, causing temperature to drop during expansion. * Equations of state: $PV^gamma = \text{constant}$, $TV^{gamma-1} = \text{constant}$, $P^{1-gamma}T^gamma = \text{constant}$, where $gamma = C_p/C_v$. * Work done: $W = \frac{P_1V_1 - P_2V_2}{gamma-1} = nC_v(T_1 - T_2)$. * P-V diagram: A steeper hyperbola than isothermal. Example: Rapid expansion/compression, or in a well-insulated container.

1
Cyclic Process:

* The system returns to its initial state. * $Delta U_{cycle} = 0$ (since $U$ is a state function). * From First Law: $Q_{net} = W_{net}$. The net heat absorbed equals the net work done. * Work done: Area enclosed by the loop on the P-V diagram. Clockwise loop means positive work (work done by system), counter-clockwise means negative work (work done on system).

Key Relations: $C_p - C_v = R$ (Mayer's relation) and $gamma = C_p/C_v$. Remember to use absolute temperature (Kelvin) in all calculations. Pay close attention to the sign conventions for $Q$ and $W$!

Prelims Revision Notes

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First Law of Thermodynamics: — $Delta U = Q - W$. This is the energy conservation principle. $Delta U$ is change in internal energy, $Q$ is heat added to the system, $W$ is work done *by* the system.
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Internal Energy of Ideal Gas: — $U = nC_vT$. Thus, $Delta U = nC_vDelta T$. It depends ONLY on temperature.
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Work Done ($W$): — $W = int P dV$. Area under P-V curve. Positive for expansion, negative for compression.
4
Isobaric Process (Constant Pressure):

* $P = \text{constant}$. * P-V diagram: Horizontal line. * Work: $W = P(V_2 - V_1)$. * Heat: $Q = nC_pDelta T$. * $Delta U = nC_vDelta T$.

1
Isochoric Process (Constant Volume):

* $V = \text{constant}$. * P-V diagram: Vertical line. * Work: $W = 0$. * Heat: $Q = nC_vDelta T$. * $Delta U = Q$.

1
Isothermal Process (Constant Temperature):

* $T = \text{constant}$. * P-V diagram: Hyperbola ($PV = \text{constant}$). Boyle's Law applies. * Internal Energy: $Delta U = 0$ (for ideal gas). * Work: $W = nRT ln(V_2/V_1) = P_1V_1 ln(V_2/V_1)$. * Heat: $Q = W$.

1
Adiabatic Process (No Heat Exchange):

* $Q = 0$. * P-V diagram: Steeper hyperbola than isothermal ($PV^gamma = \text{constant}$). Poisson's Law applies. * Internal Energy: $Delta U = -W$. * Relations: $PV^gamma = \text{constant}$, $TV^{gamma-1} = \text{constant}$, $P^{1-gamma}T^gamma = \text{constant}$. * Work: $W = \frac{P_1V_1 - P_2V_2}{gamma-1} = nC_v(T_1 - T_2)$.

1
Cyclic Process:

* System returns to initial state. * $Delta U_{cycle} = 0$. * Net Heat: $Q_{net} = W_{net}$. * Work: Area enclosed by the loop on P-V diagram. Clockwise is positive work, counter-clockwise is negative.

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Mayer's Relation: — $C_p - C_v = R$.
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Adiabatic Index: — $gamma = C_p/C_v$.
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Units: — Convert temperature to Kelvin. $1,\text{L}cdot\text{atm} = 101.3,\text{J}$. $R = 8.314,\text{J/mol}cdot\text{K}$.

Vyyuha Quick Recall

Isothermal: Temperature Constant (TC) Adiabatic: Quantity of heat Zero (QZ) Isobaric: Pressure Constant (PC) Isochoric: Volume Constant (VC)

*Think: 'TC QZ PC VC' for the constant/zero quantity in each process.*