Question

The molar heat capacity of a certain substance varies with temperature according to the given equation $C = 27.2 + \left( {4 \times {{10}^{ - 3}}} \right)T$. The heat necessary to change the temperature of $2$ mol of the substance from $300K$ to $700K$ is
(A) $3.46 \times {10^4}J$
(B) $2.33 \times {10^3}J$
(C) $3.46 \times {10^3}J$
(D) $2.33 \times {10^4}J$

Answer 1

To solve this question, we need to use the formula of the heat required in terms of the molar heat capacity. Then integrating the expression between the temperature limits given in the question, we will get the final answer.
Formula used: The formula used to solve this question is given by
$Q = nC\Delta T$, here $Q$ is the heat necessary to raise the temperature of $n$ moles of a substance having molar heat capacity of $C$ through $\Delta T$.

Complete step-by-step solution:
We know that the heat required to raise the temperature of a substance in terms of the molar heat capacity is given by
$Q = nC\Delta T$
Since the specific heat is not a constant, so we consider a small heat $dQ$ required to raise the temperature of the given substance by a small temperature of $dT$ from an initial temperature of $T$, as given below
$dQ = nCdT$ (1)
According to the question, the expression for the molar heat capacity is
$C = 27.2 + \left( {4 \times {{10}^{ - 3}}} \right)T$ (2)
Putting (2) in (1) we get
$dQ = n\left[ {27.2 + \left( {4 \times {{10}^{ - 3}}} \right)T} \right]dT$
Integrating both sides from $300K$ to $700K$, we have
$\int\limits_0^Q {dQ} = n\int\limits_{300}^{700} {\left[ {27.2 + \left( {4 \times {{10}^{ - 3}}} \right)T} \right]dT}$
$\Rightarrow \left[ Q \right]_0^Q = n\left[ {27.2T + \left( {4 \times {{10}^{ - 3}}} \right)\dfrac{{{T^2}}}{2}} \right]_{300}^{700}$
Substituting the limits, we get
$Q - 0 = n\left[ {27.2\left( {700 - 300} \right) + \left( {2 \times {{10}^{ - 3}}} \right)\left( {{{700}^2} - {{300}^2}} \right)} \right]$
$\Rightarrow Q = 11680n$
According to the question, $n = 2$. Putting this above, we get
$Q = 23360J$
$\Rightarrow Q = 2.336 \times {10^4}J \approx 2.33 \times {10^4}J$
Thus, the required value of the heat is equal to $2.33 \times {10^4}J$.

Hence, the correct answer is option 4.

Note: Do not make the mistake of substituting the values of the given temperatures into the expression of the molar heat capacity. In this way we would get the initial and the final values of the heat capacities. But the formula for the heat in terms of the molar heat capacity is valid for the constant molar heat capacity. So there was a need of integration to solve this question.