Question 8

Step 1: Convert temperatures to Kelvin. $T_1 = -73 + 273 = 200\text{ K}$ $T_2 = 27 + 273 = 300\text{ K}$

Step 2: Use the formula for heat $dQ = m C dT$. Given $m = 1\text{ kg}$ and $C = kT$. $Q = \int_{T_1}^{T_2} m C dT = \int_{200}^{300} (1)(kT) dT$ $Q = k \left[ \frac{T^2}{2} \right]_{200}^{300}$ $Q = \frac{k}{2} (300^2 - 200^2) = \frac{k}{2} (90000 - 40000)$ $Q = \frac{k}{2} (50000) = 25000k$

Step 3: Compare with the given heat $nk$. $nk = 25000k \implies n = 25000$.