Chemical reactions are invariably assocated with the transfer of energy either in the form of heat or light. In the laboratory, heat changes in physical and chemical processes are measured with an instrument called calorimeter. Heat change in the process is calculated as: q= ms `DeltaT`, s= specific heat = `c Delta T`, c= heat capacity
Heat of reaction at constant volume is measured using bomb calorimeter. `qv= Delta U=` internal energy change.
Heat of reaction at constant pressure is measured using simple or water calorimeter. `q_(p) = Delta H, q_(p) = q_(v) + P Delta V, DeltaH = DeltaU + Delta nRT`
The amount of energy released during a chemical change depnds on the physical state of reactants and products, the condition of pressure, temperature and volume at which the reaction is carried out. The variation of heat of reaction with temperature and pressure is given by Kirchhoff's equation: `(DeltaH_(2)- DeltaH_(1))/(TT_(2)-T_(1)) = DeltaC_(P)` (At constant pressure), `(DeltaU_(2)- DeltaU_(1))/(TT_(2)-T_(1)) = DeltaC_(V)` (At constant volume)
The heat capacity of bomb calorimeter (with its contents) is 500J/K. When 0.1g of `CH_(4)` was burnt in this calorimeter the temperature rose by `2^(@)C`. The value of `DeltaU` per mole will be

Options

A. `+1 kJ`
B. `-1 kJ`
C. `+160 kJ`
D. `-160 kJ`

Answer

To find the value of ΔU (change in internal energy) per mole of methane (CH₄) burnt in a bomb calorimeter, we can follow these steps:

### Step 1: Calculate the heat change (q) using the calorimeter's heat capacity.
The formula to calculate the heat change is:
\[ q = C \Delta T \]
Where:
- \( C \) = heat capacity of the calorimeter = 500 J/K
- \( \Delta T \) = change in temperature = 2 °C = 2 K (since the change in temperature in Celsius is equivalent to the change in Kelvin)

Substituting the values:
\[ q = 500 \, \text{J/K} \times 2 \, \text{K} = 1000 \, \text{J} \]

### Step 2: Convert the heat change to kilojoules.
Since 1 kJ = 1000 J:
\[ q = 1000 \, \text{J} = 1 \, \text{kJ} \]

### Step 3: Determine the number of moles of methane burnt.
The molecular mass of methane (CH₄) is approximately 16 g/mol. Given that 0.1 g of CH₄ was burnt:
\[ \text{Number of moles of CH₄} = \frac{0.1 \, \text{g}}{16 \, \text{g/mol}} = 0.00625 \, \text{mol} \]

### Step 4: Calculate ΔU per mole of methane.
We know that the heat change (q) corresponds to the change in internal energy (ΔU) for the amount of substance burnt. To find ΔU per mole, we use the formula:
\[ \Delta U \, \text{per mole} = \frac{q}{\text{number of moles}} \]
Substituting the values:
\[ \Delta U \, \text{per mole} = \frac{1 \, \text{kJ}}{0.00625 \, \text{mol}} \]

Calculating this gives:
\[ \Delta U \, \text{per mole} = 160 \, \text{kJ/mol} \]

### Conclusion:
The value of ΔU per mole of methane is:
\[ \Delta U = -160 \, \text{kJ/mol} \]
(Note: The negative sign indicates that energy is released during the combustion of methane.)

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