Change in Entropy of the Universe due to Heat Transfer

Given:
- Initial temperature of the metal, T1 = 400 K
- Final temperature of the heat bath, T2 = 100 K
- Heat capacity of the metal, C = 1000

Approach:
To find the change in entropy of the universe, we need to consider both the metal and the heat bath. The change in entropy of the universe can be calculated using the equation:

ΔS_universe = ΔS_system + ΔS_surroundings

1. Change in Entropy of the Metal (ΔS_system):
The change in entropy of the metal can be calculated using the formula:

ΔS_system = (Q/T1) - (Q/T2)

where Q is the heat transferred to the metal.

2. Change in Entropy of the Heat Bath (ΔS_surroundings):
The change in entropy of the heat bath can be calculated using the formula:

ΔS_surroundings = -Q/T2

where Q is the heat transferred from the metal to the heat bath.

Calculation:
To calculate the heat transferred, we can use the equation:

Q = C * ΔT

where ΔT is the change in temperature.

1. Change in Entropy of the Metal (ΔS_system):
ΔT = T2 - T1 = 100 K - 400 K = -300 K (temperature change is negative because the metal is losing heat)

Q = C * ΔT = 1000 * -300 = -300,000 J

ΔS_system = (-300,000 J / 400 K) - (-300,000 J / 100 K) = -750 J/K

2. Change in Entropy of the Heat Bath (ΔS_surroundings):
Q = -300,000 J (negative sign denotes heat transferred from the metal to the heat bath)

ΔS_surroundings = -Q/T2 = -(-300,000 J) / 100 K = 3,000 J/K

Final Result:
ΔS_universe = ΔS_system + ΔS_surroundings = -750 J/K + 3,000 J/K = 2,250 J/K

Therefore, the change in entropy of the universe due to the heat transfer is 2,250 J/K.