Question:

A copper rod of length L is initially at a temperature T1 and is in thermal equilibrium with its surroundings at a temperature T2. The rod is uniformly heated from one end and reaches a new equilibrium temperature T3. The coefficient of linear expansion for copper is α.

a) Derive an expression for the change in length of the copper rod when it is heated. (5 points)

b) Calculate the change in length of the copper rod, assuming its initial length is 2.0 meters, the temperature difference between T3 and T1 is 200°C, and the coefficient of linear expansion for copper is α = 1.7 × 10^-5 °C^-1. (7 points)

c) Calculate the amount of heat transferred to the copper rod during this process, given that its specific heat capacity is c = 0.385 J/g°C. Assume the mass of the rod is 500 grams. (8 points)

Answer:

a) To find the change in length of the copper rod when it is heated, we can use the equation for linear expansion:

ΔL = α * L0 * ΔT

Where: ΔL is the change in length α is the coefficient of linear expansion L0 is the initial length of the rod ΔT is the change in temperature

b) Substituting the values into the equation derived in part (a):

ΔL = α * L0 * ΔT ΔL = (1.7 × 10^-5 °C^-1) * (2.0 meters) * (200 °C) ΔL = 0.0068 meters

Therefore, the change in length of the copper rod is 0.0068 meters.

c) The amount of heat transferred to the copper rod can be calculated using the equation:

Q = m * c * ΔT

Where: Q is the amount of heat transferred m is the mass of the rod c is the specific heat capacity ΔT is the change in temperature

Substituting the values:

Q = (500 grams) * (0.385 J/g°C) * (200 °C) Q = 38,500 J

Therefore, the amount of heat transferred to the copper rod during this process is 38,500 J.