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A sample of one mole of a diatomic ideal gas initially occupies a volume of 8 liters at a temperature of $400 \mathrm{~K}.$ The gas is allowed to expand isothermally to a final volume of 20 liters. Calculate:

Option: 1
$\mathrm{2078.5 L-atm , 2110.5 L-atm}$

Option: 2
$\mathrm{2078.5 L-atm , -2110.5 L-atm}$

Option: 3
$\mathrm{-2078.5 L-atm , 2110.5 L-atm}$

Option: 4
$\mathrm{-2078.5 L-atm, -2110.5 L-atm}$

Answers (1)

best_answer

(A) Change in internal energy in L-atm

Step 1: Use the ideal gas law to relate the initial and final conditions.

$\mathrm{ P V=\eta R T }$
Step 2: Calculate the initial and final pressures using the ideal gas law.

$\mathrm{ P_1=\frac{\eta R}{V_1} T_1 }$

$\mathrm{ P_2=\frac{n R}{V_2} T_2 }$

Step 3: Calculate the change in internal energy using the formula for ideal gases.
$\mathrm{ \Delta U=\frac{5}{2} n R\left(T_2-T_1\right) }$
Step 4: Substitute the given values and calculate the change in internal energy.

$\mathrm{ \Delta U=\frac{5}{2} \times(1 \mathrm{~mol}) \times(8.314 \mathrm{~L}-\mathrm{atm} / \mathrm{mol}-\mathrm{K}) \times(400 \mathrm{~K}-300 \mathrm{~K}) }$
Calculating the numerical value of the change in internal energy:

$\mathrm{ \Delta U=2078.5 \mathrm{~L} \text {-atm } }$

(B) Change in heat enthalpy in L-atm
Step 1: Use the formula for change in heat enthalpy $\mathrm{(\Delta H)}$ for an ideal gas at constant pressure.

$\mathrm{ \Delta H=\Delta U+P \Delta V }$

Step 2: Calculate the change in volume $\mathrm{ (\Delta V) }$ using the initial and final volumes.

$\mathrm{ \Delta V=V_2-V_1 }$

Step 3: Calculate the change in heat enthalpy using the calculated values.

$\mathrm{ \Delta H=\Delta U+P_2 \Delta V }$

Step 4: Substitute the given values and calculate the change in heat enthalpy.

$\mathrm{ \Delta H=2078.5 \mathrm{~L}-\mathrm{atm}+(2 \mathrm{~atm}) \times(20 \mathrm{~L}-8 \mathrm{~L}) }$

Calculating the numerical value of the change in heat enthalpy:

$\mathrm{ \Delta H=2110.5 \mathrm{~L}-\mathrm{atm} }$

So, correct option is 1

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