First Law of Thermodynamics — Revision Notes

⚡ 30-Second Revision

First Law — $\Delta U = q + w$ (Conservation of Energy)
Internal Energy ($U$) — State function. For ideal gas, $U = f(T)$ only. $\Delta U = nC_v\Delta T$.
Heat ($q$) — Path function. $+q$ (absorbed), $-q$ (released).
Work ($w$) — Path function. $+w$ (on system), $-w$ (by system).
PV Work — $w = -P_{ext}\Delta V$.
Reversible Isothermal Work (Ideal Gas) — $w = -nRT \ln(V_2/V_1) = -nRT \ln(P_1/P_2)$.
Isochoric Process ($\Delta V = 0$) — $w = 0 \implies \Delta U = q_v$.
Isobaric Process ($P = ext{constant}$) — $w = -P\Delta V \implies \Delta H = q_p$. Enthalpy $H = U + PV$.
Isothermal Process ($\Delta T = 0$) — For ideal gas, $\Delta U = 0 \implies q = -w$.
Adiabatic Process ($q = 0$) — $\Delta U = w_{ad}$.
Cyclic Process — $\Delta U = 0 \implies q_{cycle} = -w_{cycle}$.
Mayer's Relation (Ideal Gas) — $C_p - C_v = R$.
Conversion — $1,\text{L atm} = 101.3,\text{J}$.

2-Minute Revision

The First Law of Thermodynamics is the principle of energy conservation: energy can't be created or destroyed, only transformed. Its mathematical form is $Delta U = q + w$, where $Delta U$ is the change in internal energy, $q$ is heat, and $w$ is work.

Internal energy ($U$) is a state function, meaning its change depends only on the initial and final states. For an ideal gas, $U$ depends solely on temperature. Heat ($q$) and work ($w$) are path functions, their values depend on the specific process.

Crucial sign conventions: $q$ is positive if heat is absorbed by the system, negative if released. $w$ is positive if work is done *on* the system (e.g., compression), negative if work is done *by* the system (e.g., expansion). For PV work, $w = -P_{ext}Delta V$.

Different processes simplify the First Law: In an isochoric (constant volume) process, $w=0$, so $Delta U = q_v$. In an isobaric (constant pressure) process, $Delta H = q_p$, where enthalpy ($H=U+PV$) is introduced.

In an isothermal (constant temperature) process, for an ideal gas, $Delta U=0$, so $q=-w$. In an adiabatic (no heat exchange) process, $q=0$, so $Delta U = w_{ad}$. For a cyclic process, $Delta U=0$, so $q_{cycle}=-w_{cycle}$.

Remember Mayer's relation for ideal gases: $C_p - C_v = R$. Always pay attention to units and temperature in Kelvin.

5-Minute Revision

The First Law of Thermodynamics is a fundamental principle stating that energy is conserved. It's expressed as $Delta U = q + w$, where $Delta U$ is the change in the system's internal energy, $q$ is the heat exchanged, and $w$ is the work done. Internal energy ($U$) is the total energy within the system at the molecular level; it's a state function, meaning $Delta U$ depends only on the initial and final states. For an ideal gas, $U$ is solely dependent on temperature.

**Heat ($q$)** is energy transfer due to temperature difference. By convention, $q > 0$ when absorbed by the system, $q < 0$ when released. **Work ($w$)** is energy transfer not due to temperature difference, primarily PV work in chemistry.

By convention, $w > 0$ when done *on* the system (compression), $w < 0$ when done *by* the system (expansion). The formula for work against constant external pressure is $w = -P_{ext}Delta V$. For a reversible isothermal expansion of an ideal gas, $w = -nRT ln(V_2/V_1)$.

Both $q$ and $w$ are path functions.

Let's review the First Law's application to key processes:

1
Isochoric (Constant Volume, $Delta V = 0$) — Since $w = -P_{ext}Delta V = 0$, the First Law becomes $Delta U = q_v$. All heat goes into internal energy.
2
Isobaric (Constant Pressure, $P = ext{constant}$) — Here, $w = -PDelta V$. The heat exchanged at constant pressure is defined as the change in enthalpy, $Delta H = q_p$. Enthalpy ($H = U + PV$) is also a state function. This is crucial for thermochemistry.
3
Isothermal (Constant Temperature, $Delta T = 0$) — For an ideal gas, $Delta U = 0$. Thus, $q = -w$. Heat absorbed is entirely converted to work done by the system, or vice-versa.
4
Adiabatic (No Heat Exchange, $q = 0$) — The First Law simplifies to $Delta U = w_{ad}$. Work done directly changes internal energy.
5
Cyclic Process — The system returns to its initial state, so $Delta U_{cycle} = 0$. This implies $q_{cycle} = -w_{cycle}$.

Heat Capacities: Molar heat capacity at constant volume ($C_v$) relates to $Delta U = nC_vDelta T$. Molar heat capacity at constant pressure ($C_p$) relates to $Delta H = nC_pDelta T$. For ideal gases, Mayer's relation states $C_p - C_v = R$. Remember to convert units (e.g., L atm to J using $1,\text{L atm} = 101.3,\text{J}$) and use temperature in Kelvin. Mastering these concepts and their associated formulas, along with strict adherence to sign conventions, is key to solving NEET problems.

Prelims Revision Notes

First Law of Thermodynamics: Key Facts for NEET

1. Fundamental Principle:

Law of Conservation of Energy: — Energy cannot be created or destroyed; it only changes forms.
Mathematical Form: — $\Delta U = q + w$

* $\Delta U$: Change in internal energy of the system. * $q$: Heat exchanged between system and surroundings. * $w$: Work done on or by the system.

2. Key Terms & Conventions:

**Internal Energy ($U$):**

* Total energy stored within a system (kinetic + potential energy of molecules). * State Function: Depends only on the initial and final states, not the path. $\Delta U_{cycle} = 0$. * For an ideal gas, $U$ depends only on temperature ($T$). Thus, for isothermal process of ideal gas, $\Delta U = 0$.

**Heat ($q$):**

* Path Function: Depends on the process path. * Sign Convention: * $q > 0$: Heat absorbed by the system (endothermic). * $q < 0$: Heat released by the system (exothermic).

**Work ($w$):**

* Path Function: Depends on the process path. * Sign Convention (IUPAC, Chemistry): * $w > 0$: Work done *on* the system by surroundings (e.g., compression). * $w < 0$: Work done *by* the system on surroundings (e.g., expansion). * PV Work: $w = -P_{ext}\Delta V$ (for constant external pressure). * $\Delta V = V_{final} - V_{initial}$. * If $\Delta V > 0$ (expansion), $w < 0$. * If $\Delta V < 0$ (compression), $w > 0$.

**Enthalpy ($H$):**

* Defined as $H = U + PV$. * State Function. * At constant pressure: $\Delta H = q_p$. * **Relation to $\Delta U$ for gaseous reactions:** $\Delta H = \Delta U + \Delta n_g RT$, where $\Delta n_g = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants})$.

3. Thermodynamic Processes & First Law Simplifications:

**Isochoric Process (Constant Volume, $\Delta V = 0$):**

* $w = 0$. * $\Delta U = q_v$ (heat at constant volume).

**Isobaric Process (Constant Pressure, $P = \text{constant}$):**

* $w = -P\Delta V$. * $\Delta U = q_p - P\Delta V \implies \Delta H = q_p$.

**Isothermal Process (Constant Temperature, $\Delta T = 0$):**

* For ideal gas: $\Delta U = 0$. * Therefore, $q = -w$. * Reversible Isothermal Work (Ideal Gas): $w = -nRT \ln\left(\frac{V_2}{V_1}\right) = -nRT \ln\left(\frac{P_1}{P_2}\right)$.

**Adiabatic Process (No Heat Exchange, $q = 0$):**

* $\Delta U = w_{ad}$. * System is insulated or process is very rapid.

Cyclic Process (Returns to initial state):

* $\Delta U_{cycle} = 0$. * Therefore, $q_{cycle} = -w_{cycle}$.

4. Heat Capacities:

Molar Heat Capacity at Constant Volume ($C_v$): — $q_v = nC_v\Delta T \implies \Delta U = nC_v\Delta T$.
Molar Heat Capacity at Constant Pressure ($C_p$): — $q_p = nC_p\Delta T \implies \Delta H = nC_p\Delta T$.
Mayer's Relation (for ideal gas): — $C_p - C_v = R$.
Adiabatic Index ($\gamma$): — $\gamma = C_p/C_v$.

5. Units & Conversions:

Energy: Joules (J), kilojoules (kJ).
$1,\text{L atm} = 101.3,\text{J}$.
Temperature: Always use Kelvin (K). $T(\text{K}) = T(^circ\text{C}) + 273.15$.
Gas Constant ($R$): $8.314,\text{J mol}^{-1}\text{K}^{-1}$ or $0.0821,\text{L atm mol}^{-1}\text{K}^{-1}$.

Common Mistakes to Avoid:

Incorrect sign conventions for $q$ and $w$.
Forgetting to convert units (L atm to J, $^circ\text{C}$ to K).
Confusing state functions with path functions.
Misapplying formulas for different processes (e.g., using isothermal work formula for adiabatic process).

Vyyuha Quick Recall

To remember the First Law and its signs: 'Q-W-U'

Queer Work Understood: $\Delta U = q + w$
Queer (Heat): Quickly Increases (positive for absorbed), Out (negative for released).
Work: When On (positive for on system), By (negative for by system).

For processes: 'I-A-I-A-C' (Isothermal, Adiabatic, Isochoric, Isobaric, Cyclic)

Isothermal: Temperature Constant ($\Delta T=0 \implies \Delta U=0 \implies q=-w$ for ideal gas).
Adiabatic: Quiet (No heat, $q=0 \implies \Delta U=w$).
Isochoric: Volume Constant ($\Delta V=0 \implies w=0 \implies \Delta U=q_v$).
Isobaric: Pressure Constant ($\Delta H=q_p$).
Cyclic: U-turn (Back to start, $\Delta U=0 \implies q=-w$).