Internal Energy — Revision Notes

⚡ 30-Second Revision

Internal Energy ($U$): — Total microscopic energy of a system. State function.
First Law of Thermodynamics: — $\Delta U = q + w$.
Sign Conventions:

* $q > 0$: Heat absorbed by system. * $q < 0$: Heat released by system. * $w > 0$: Work done on system (compression). * $w < 0$: Work done by system (expansion).

PV Work: — $w = -P_{ext} \Delta V$.
Isochoric Process ($ \Delta V = 0 $): — $w = 0 \implies \Delta U = q_v$.
Ideal Gas Internal Energy: — $U = f(T)$ only.
Ideal Gas $ \Delta U $: — $\Delta U = n C_v \Delta T$.
$C_v$ for Ideal Gases:**

* Monatomic: $C_v = \frac{3}{2}R$. * Diatomic: $C_v = \frac{5}{2}R$ (at moderate T).

Isothermal Process (Ideal Gas): — $\Delta T = 0 \implies \Delta U = 0$.

2-Minute Revision

Internal energy ($U$) is the sum of all kinetic and potential energies of particles within a system, excluding bulk motion. It's a state function, meaning its value depends only on the system's current state (T, P, V) and not the path taken.

The First Law of Thermodynamics, $\Delta U = q + w$, is central, stating that the change in internal energy equals heat absorbed by the system plus work done on the system. Remember the crucial sign conventions: $q$ is positive for absorption, negative for release; $w$ is positive for work on the system, negative for work by the system.

For pressure-volume work, $w = -P_{ext} \Delta V$. In an isochoric (constant volume) process, $w=0$, so $\Delta U = q_v$. For ideal gases, internal energy depends solely on temperature, and $\Delta U = n C_v \Delta T$.

For monatomic ideal gases, $C_v = \frac{3}{2}R$, and for diatomic, $C_v = \frac{5}{2}R$ (at moderate temperatures). Crucially, for an ideal gas undergoing an isothermal (constant temperature) process, $\Delta U = 0$.

5-Minute Revision

Internal energy ($U$) is a fundamental thermodynamic property representing the total energy stored within a system at the microscopic level. This includes the kinetic energies of molecular translation, rotation, and vibration, as well as potential energies from intermolecular forces and chemical bonds.

It's vital to remember that $U$ is a state function, meaning its value is determined solely by the system's current state (e.g., temperature, pressure, volume) and is independent of the path taken to reach that state.

We cannot measure the absolute value of $U$, only its changes ($\Delta U$).

The First Law of Thermodynamics is the cornerstone of internal energy calculations: $\Delta U = q + w$. Here, $\Delta U$ is the change in internal energy of the system, $q$ is the heat transferred to the system, and $w$ is the work done on the system.

Strict adherence to sign conventions is critical: $q$ is positive if heat is absorbed by the system (endothermic), negative if released (exothermic). $w$ is positive if work is done on the system (e.

g., compression), negative if work is done by the system (e.g., expansion). For pressure-volume work against a constant external pressure, $w = -P_{ext} \Delta V$.

Special Cases and Ideal Gases:

1
Isochoric Process (Constant Volume): — If $\Delta V = 0$, then $w = -P_{ext} \Delta V = 0$. The First Law simplifies to $\Delta U = q_v$. This means the heat exchanged at constant volume directly equals the change in internal energy, as measured in a bomb calorimeter.
2
Ideal Gases: — For an ideal gas, internal energy is considered to be a function of temperature only. This is because ideal gas particles have no intermolecular forces (no potential energy component from interactions) and negligible volume. Therefore, $\Delta U = n C_v \Delta T$, where $n$ is moles, $C_v$ is molar heat capacity at constant volume, and $\Delta T$ is temperature change. For monatomic ideal gases, $C_v = \frac{3}{2}R$, and for diatomic ideal gases (at moderate temperatures), $C_v = \frac{5}{2}R$.
3
Isothermal Process (Constant Temperature) for Ideal Gas: — Since $\Delta T = 0$, for an ideal gas, $\Delta U = n C_v (0) = 0$. In this case, from the First Law, $q = -w$, meaning any heat absorbed is entirely converted to work done by the system, and vice versa.

Example: $2$ moles of an ideal diatomic gas expand isothermally against a constant external pressure of $1\,\text{atm}$ from $10\,\text{L}$ to $20\,\text{L}$ at $300\,\text{K}$. Calculate $\Delta U$.

Since it's an ideal gas and the process is isothermal ($\Delta T = 0$), $\Delta U = 0$.
Even though work is done and heat is exchanged ($w = -P_{ext} \Delta V = -1\,\text{atm} \times (20-10),\text{L} = -10\,\text{L atm} = -1013\,\text{J}$), and $q = -w = +1013\,\text{J}$, the net change in internal energy is zero because the temperature of the ideal gas remains constant.

Prelims Revision Notes

Internal energy ($U$) is a crucial concept in NEET chemistry, representing the total energy within a system at a microscopic level. It's a state function, meaning its value depends only on the system's current state (T, P, V) and not the path taken. We measure changes in internal energy ($\Delta U$), not absolute values.

First Law of Thermodynamics: $\Delta U = q + w$.

$\Delta U$: Change in internal energy of the system.
$q$: Heat transferred. Positive if absorbed by system, negative if released.
$w$: Work done. Positive if done on system (compression), negative if done by system (expansion).

Work (PV-Work): For expansion/compression against constant external pressure, $w = -P_{ext} \Delta V$. Remember $1\,\text{L atm} \approx 101.3\,\text{J}$.

Specific Processes:

1
Isochoric (Constant Volume, $ \Delta V = 0 $): — Since $\Delta V = 0$, $w = 0$. Thus, $\Delta U = q_v$. Bomb calorimetry measures $q_v$, hence $\Delta U$.
2
**Isothermal (Constant Temperature, $\Delta T = 0$):**

* For ideal gases: $U$ depends only on $T$. So, if $\Delta T = 0$, then $\Delta U = 0$. In this case, $q = -w$. * For real gases/liquids/solids: $\Delta U$ may not be zero even if $\Delta T = 0$ due to changes in intermolecular potential energy (e.g., phase changes).

1
Adiabatic (No Heat Exchange, $q = 0 $): — $\Delta U = w$. Any change in internal energy is solely due to work done.

Internal Energy of Ideal Gases:

Depends only on temperature. $U = f(T)$.
$\Delta U = n C_v \Delta T$.
**Molar Heat Capacity at Constant Volume ($C_v$):**

* Monatomic ideal gas (He, Ne, Ar): $C_v = \frac{3}{2}R$. * Diatomic ideal gas (N$_2$, O$_2$, H$_2$): $C_v = \frac{5}{2}R$ (at moderate temperatures, considering translational and rotational modes). * Polyatomic ideal gas: $C_v = \frac{3}{2}R + R_{rot} + R_{vib}$ (more complex, often given or derived).

Key Distinctions:

Internal Energy vs. Heat: — $U$ is a property of the system (state function), $q$ is energy in transit (path function).
Internal Energy vs. Enthalpy: — $\Delta U$ is $q_v$, $\Delta H$ is $q_p$. Relationship: $\Delta H = \Delta U + \Delta n_g RT$ for gaseous reactions.

Common Mistakes to Avoid:

Incorrect sign conventions for $q$ and $w$.
Forgetting to convert temperature to Kelvin.
Using $C_p$ instead of $C_v$ for $\Delta U$ calculations.
Confusing ideal gas behavior with real gas behavior for $\Delta U$ in isothermal processes.

Vyyuha Quick Recall

Understand Quickly Work: $\Delta U = q + w$.

Understand: $\Delta U$ is the change in Unique (internal) energy.
Quickly: $q$ is Quantity of heat (positive if absorbed, negative if released).
Work: $w$ is Work (positive if done on system, negative if done by system).