Step-by-step Solution:

1. Identify Given Information:

   • Mass of the substance, $m = 1 \mathrm{~kg}$.
   • Temperature-dependent specific heat capacity, $C = kT$, where $T$ is the absolute temperature in Kelvin.
   • Initial temperature, $T_{initial}(^{\circ} \mathrm{C}) = -73^{\circ} \mathrm{C}$.
   • Final temperature, $T_{final}(^{\circ} \mathrm{C}) = 27^{\circ} \mathrm{C}$.
   • Total heat required, $Q = nk$.
   • Conversion factor: $T(\mathrm{K}) = T(^{\circ} \mathrm{C}) + 273$.

2. Convert Temperatures to Kelvin: The formula for specific heat capacity is given in terms of absolute temperature (Kelvin). Therefore, we must convert the given temperatures from Celsius to Kelvin.

   • Initial temperature, $T_1 = -73 + 273 = 200 \mathrm{~K}$.
   • Final temperature, $T_2 = 27 + 273 = 300 \mathrm{~K}$.

3. Set up the Heat Calculation: When the specific heat capacity is a function of temperature, the heat required ($Q$) to raise the temperature of a mass ($m$) from $T_1$ to $T_2$ is given by the integral: $Q = \int_{T_1}^{T_2} mC(T) \, dT$

4. Substitute the Given Specific Heat Capacity: We are given $C(T) = kT$. Substituting this into the integral: $Q = \int_{T_1}^{T_2} m(kT) \, dT$

5. Evaluate the Integral: Since $m$ and $k$ are constants, we can take them out of the integral: $Q = mk \int_{T_1}^{T_2} T \, dT$ The integral of $T$ with respect to $T$ is $\frac{T^2}{2}$. Evaluating this from $T_1$ to $T_2$: $Q = mk \left[ \frac{T^2}{2} \right]_{T_1}^{T_2}$ $Q = mk \left( \frac{T_2^2}{2} - \frac{T_1^2}{2} \right)$ $Q = \frac{mk}{2} (T_2^2 - T_1^2)$

6. Substitute Numerical Values: Now, we plug in the values for $m$, $T_1$, and $T_2$:

   • $m = 1 \mathrm{~kg}$
   • $T_1 = 200 \mathrm{~K}$
   • $T_2 = 300 \mathrm{~K}$

   $Q = \frac{(1)k}{2} ( (300)^2 - (200)^2 )$ $Q = \frac{k}{2} ( 90000 - 40000 )$ $Q = \frac{k}{2} ( 50000 )$ $Q = 25000 k$

7. Determine the Value of n: The problem states that the heat required is $Q = nk$. Comparing this with our calculated result: $nk = 25000 k$ Dividing both sides by $k$, we get: $n = 25000$

Final Answer:

The value of $n$ is 25000.