Measurement of ??U and ??H — Revision Notes

⚡ 30-Second Revision

Internal Energy Change ($\Delta U$): — Heat at constant volume ($q_V$). Measured by Bomb Calorimeter. \n * Formula: $\Delta U = -C_{calorimeter} \times \Delta T$ \n- **Enthalpy Change ($\Delta H$):** Heat at constant pressure ($q_P$). Measured by Coffee-Cup Calorimeter. \n * Formula: $\Delta H = -(m_{solution} \times c_{solution} \times \Delta T)$ \n- Relationship: $\Delta H = \Delta U + \Delta n_g RT$ \n * $\Delta n_g = (\text{moles of gaseous products}) - (\text{moles of gaseous reactants})$ \n * $R = 8.314\text{ J/mol\cdot K}$ (use in Joules, convert to kJ if needed) \n * $T$ must be in Kelvin ($T(\text{K}) = T(\text{\textdegree C}) + 273$) \n- Sign Convention: Exothermic (heat released) $\rightarrow \Delta U, \Delta H < 0$. Endothermic (heat absorbed) $\rightarrow \Delta U, \Delta H > 0$.

2-Minute Revision

For NEET, understanding the measurement of $\Delta U$ and $\Delta H$ is crucial. $\Delta U$ represents the change in internal energy, which is the heat exchanged at constant volume ($q_V$). It's measured using a bomb calorimeter, a rigid, sealed device.

The calculation involves the calorimeter's heat capacity ($C_{calorimeter}$) and the temperature change ($\Delta T$): $\Delta U = -C_{calorimeter} \times \Delta T$. The negative sign indicates that heat released by the reaction is absorbed by the calorimeter.

\n\n$\Delta H$ represents the change in enthalpy, which is the heat exchanged at constant pressure ($q_P$). This is typically measured using a coffee-cup calorimeter, a simpler device suitable for reactions in solution.

The calculation uses the mass of the solution ($m_{solution}$), its specific heat capacity ($c_{solution}$), and the temperature change: $\Delta H = -(m_{solution} \times c_{solution} \times \Delta T)$.

\n\nThese two quantities are related by the equation $\Delta H = \Delta U + \Delta n_g RT$. Here, $\Delta n_g$ is the change in moles of gaseous species (products minus reactants), $R$ is the gas constant ($8.

314\text{ J/mol\cdot K}$), and$T$ is the absolute temperature in Kelvin. Remember to use consistent units and pay close attention to the sign conventions for exothermic and endothermic reactions.

5-Minute Revision

A thorough understanding of $\Delta U$ and $\Delta H$ measurement is vital for NEET. $\Delta U$, the change in internal energy, quantifies heat exchange at constant volume ($q_V$), meaning no pressure-volume work is done.

This is experimentally determined using a bomb calorimeter. A known mass of substance is combusted in a sealed steel bomb, immersed in water. The heat released by the reaction ($q_{reaction}$) is absorbed by the calorimeter system, causing a temperature rise ($\Delta T$).

The calculation is $\Delta U = q_{reaction} = -C_{calorimeter} \times \Delta T$. $C_{calorimeter}$ is the total heat capacity of the calorimeter, including the bomb and water, usually determined by calibration.

\n\n$\Delta H$, the change in enthalpy, quantifies heat exchange at constant pressure ($q_P$), which is common for reactions in open vessels. It accounts for both internal energy change and any pressure-volume work.

It's measured using a coffee-cup calorimeter, typically nested Styrofoam cups. Reactants are mixed in solution, and the temperature change of the solution is monitored. The calculation is $\Delta H = q_{reaction} = -(m_{solution} \times c_{solution} \times \Delta T)$.

Here, $m_{solution}$ is the total mass of the solution, and $c_{solution}$ is its specific heat capacity (often approximated as water's). \n\nCrucially, $\Delta U$ and $\Delta H$ are related by $\Delta H = \Delta U + \Delta n_g RT$.

$\Delta n_g$ is the change in the number of moles of gaseous products minus gaseous reactants. For example, for $2H_2(g) + O_2(g) \rightarrow 2H_2O(l)$, $\Delta n_g = 0 - (2+1) = -3$. If $\Delta n_g = 0$ (e.

g., $H_2(g) + Cl_2(g) \rightarrow 2HCl(g)$), then $\Delta H = \Delta U$. Remember to use $R = 8.314\text{ J/mol\cdot K}$ and convert temperature to Kelvin. Always ensure consistent units (J vs. kJ) and correct sign conventions: negative for exothermic (heat released, temperature increase), positive for endothermic (heat absorbed, temperature decrease).

\n\nWorked Example: If $\Delta U = -100\text{ kJ}$ for a reaction where $\Delta n_g = +2$ at $300\text{ K}$, then $\Delta H = -100\text{ kJ} + (2\text{ mol} \times 8.314\text{ J/mol\cdot K} \times 300\text{ K}) = -100\text{ kJ} + 4988.

4\text{ J} = -100\text{ kJ} + 4.9884\text{ kJ} = -95.0116\text{ kJ}$.

Prelims Revision Notes

**1. Internal Energy Change ($\Delta U$):** \n* Definition: Heat exchanged at constant volume ($q_V$). \n* Measurement: Bomb calorimeter. \n* Principle: Rigid, sealed container ensures $\Delta V = 0$, so no $P\Delta V$ work.

$\Delta U = q_V$. \n* Formula: $q_{reaction} = -q_{calorimeter} = -C_{calorimeter} \times \Delta T$. \n* **$C_{calorimeter}$:** Heat capacity of the entire calorimeter system (bomb, water, stirrer).

Determined by calibration. \n* Units: Joules (J) or kilojoules (kJ). \n\n**2. Enthalpy Change ($\Delta H$):** \n* Definition: Heat exchanged at constant pressure ($q_P$). \n* Measurement: Coffee-cup calorimeter.

\n* Principle: Open to atmosphere, so constant pressure. $\Delta H = q_P$. \n* Formula: $q_{reaction} = -q_{solution} = -(m_{solution} \times c_{solution} \times \Delta T)$. \n* **$m_{solution}$:** Total mass of the solution.

\n* **$c_{solution}$:** Specific heat capacity of the solution (often approximated as $4.184\text{ J/g\cdot \textdegree C}$ for water). \n* Units: Joules (J) or kilojoules (kJ). \n\n**3. Relationship between $\Delta U$ and $\Delta H$:** \n* Formula: $\Delta H = \Delta U + \Delta n_g RT$.

\n* **$\Delta n_g$:** Change in moles of gaseous species. $\Delta n_g = (\sum n_{g,products}) - (\sum n_{g,reactants})$. Only count gases! \n* **$R$:** Ideal gas constant. Use $8.314\text{ J/mol\cdot K}$ (ensure unit consistency with $\Delta U, \Delta H$).

\n* **$T$:** Absolute temperature in Kelvin ($T(\text{K}) = T(\text{\textdegree C}) + 273.15$). \n* Special Case: If $\Delta n_g = 0$, then $\Delta H = \Delta U$. This happens for reactions with no gaseous reactants/products, or when moles of gaseous reactants equal moles of gaseous products.

\n\n4. Key Points for NEET: \n* Sign Convention: Exothermic reactions (heat released, temperature increase) have negative $\Delta U$ and $\Delta H$. Endothermic reactions (heat absorbed, temperature decrease) have positive $\Delta U$ and $\Delta H$.

\n* Unit Conversion: Always convert J to kJ or vice-versa to match other values. Convert $\text{\textdegree C}$ to K for $T$ in $RT$ term. \n* Common Traps: Incorrect $\Delta n_g$ calculation (including non-gaseous species), sign errors, unit inconsistencies.

\n* Applications: Bomb calorimeter for combustion; coffee-cup for solution reactions (neutralization, dissolution).

Vyyuha Quick Recall

Bomb Under Constant Volume, Coffee Heats Pressure. \n\n* Bomb: Bomb Calorimeter measures U: $\Delta U$ (Internal Energy) under Constant Volume. \n* Coffee: Coffee-Cup Calorimeter measures H: $\Delta H$ (Enthalpy) under Pressure (Constant Pressure).

\n\nAnd for the relationship: Happy Uncles Never Really Tire. \n* Happy ($\Delta H$) = Uncles ($\Delta U$) + Never ($\Delta n_g$) Really ($R$) Tire ($T$). (Remember $\Delta n_g$ is for gases only!