The temperature dependence of entropy can be determined by considering the relationship between entropy and heat capacity. Entropy is a measure of the disorder or randomness of a system, while heat capacity measures the amount of heat energy required to raise the temperature of a system.

Heat capacity at constant volume (Cv) is defined as the amount of heat required to raise the temperature of a substance by one degree Celsius at constant volume. In this case, the heat capacity of the metal is given by the equation Cv = aT + bT³, where T is the temperature and a and b are constants.

To find the temperature dependence of entropy, we need to recall the relationship between entropy (S) and heat capacity (C). The entropy change (ΔS) of a system is related to the heat transfer (Q) and temperature (T) by the equation ΔS = Q/T.

Now, let's analyze the given equation for heat capacity at constant volume (Cv = aT + bT³) to determine its relationship with entropy.

1. Constant volume heat capacity equation:
- Cv = aT + bT³

2. Entropy equation:
- ΔS = Q/T

3. Relationship between entropy and heat capacity:
- ΔS = Q/T = Cv/T

From the above equations, we can see that the entropy change (ΔS) is equal to the heat capacity at constant volume (Cv) divided by the temperature (T). Therefore, the entropy change depends on the temperature.

Let's further analyze the given equation for heat capacity (Cv = aT + bT³) to determine the temperature dependence of entropy:

1. Cv = aT + bT³

2. ΔS = Cv/T = (aT + bT³)/T = a + bT²

The final equation for the temperature dependence of entropy is ΔS = a + bT². This equation shows that the entropy change is independent of the linear term (a) and depends on the square of the temperature (T²) multiplied by a constant (b).

In conclusion, the temperature dependence of entropy for the metal, given the heat capacity equation Cv = aT + bT³, is given by the equation ΔS = a + bT². The entropy change depends on the square of the temperature and is independent of the linear term.