Calculate Heat Required for Temperature Dependent Specific Heat Capacity

To find the value of $n$, we start by expressing the infinitesimal amount of heat $dQ$ required to raise the temperature of a mass $m$ of the substance by $dT$:

$dQ = m C \, dT$

Given that the mass of the substance is $m = 1\text{ kg}$ and its specific heat capacity is $C = kT$, the equation simplifies to:

$dQ = (1\text{ kg}) \cdot (kT) \, dT = kT \, dT$

Next, we convert the initial and final temperatures from degrees Celsius to Kelvin using $0\text{ K} = -273^\circ\text{C}$:

$T_1 = -73^\circ\text{C} = -73 + 273 = 200\text{ K}$ $T_2 = 27^\circ\text{C} = 27 + 273 = 300\text{ K}$

The total heat $Q$ required to raise the temperature from $T_1$ to $T_2$ is obtained by integrating $dQ$ over the temperature range:

$Q = \int_{T_1}^{T_2} kT \, dT = k \left[ \frac{T^2}{2} \right]_{200}^{300}$

Substitute the limits of integration:

$Q = \frac{k}{2} \left( 300^2 - 200^2 \right)$ $Q = \frac{k}{2} \left( 90000 - 40000 \right)$ $Q = \frac{k}{2} \times 50000 = 25000k$

We are given that the heat required is equal to $nk$:

$Q = nk = 25000k$

Comparing both sides gives:

$n = 25000$