The internal energy of a system is given by the formula E = bs^3/VN, where b is a constant and other symbols have their usual meaning.


To find the temperature of the system, we need to use the relationship between temperature and internal energy. In thermodynamics, the temperature is defined as the rate of change of internal energy with respect to entropy (T = dE/dS), where T represents temperature, E represents internal energy, and S represents entropy.

To find the temperature, we need to differentiate the given equation for internal energy with respect to entropy. However, since the equation only contains variables s, V, and N, we need to express entropy S in terms of these variables.


Entropy is given by the formula S = k ln(W), where k is Boltzmann's constant and W represents the number of microstates accessible to the system.

In this case, since we have N particles in the system, each with s possible states, the total number of microstates W is given by W = s^N.

Substituting this into the equation for entropy, we have S = k ln(s^N) = Nk ln(s).


Now, we can differentiate the equation for internal energy with respect to entropy:

dE/dS = d(bs^3/VN)/d(Nk ln(s))
= d(bs^3/VN)/(dNk ln(s))/ds

Using the chain rule, we can simplify this expression:

dE/dS = bs^3/VN * 1/(Nk ln(s)) * d(Nk ln(s))/ds
= bs^3/VN * 1/(Nk ln(s)) * (dNk/ds * ln(s) + Nk * 1/s)

Simplifying further, we have:

dE/dS = bs^3/VN * 1/(Nk ln(s)) * (k * ln(s) + Nk/s)
= b * s^3/(VN ln(s)) * (k * ln(s) + Nk/s)
= b * (k * s^3 * ln(s) + Nk^2)/(VN ln(s) * s)


Now that we have the expression for dE/dS, we can equate it to the temperature T:

T = dE/dS = b * (k * s^3 * ln(s) + Nk^2)/(VN ln(s) * s)

Comparing this expression with the answer choices, we find that the correct option is (3) bs^3/V^3N.


The correct expression for the temperature of the system is given by T = bs^3/V^3N. This result is obtained by differentiating the equation for internal energy with respect to entropy and equating it to the temperature. The derivation involves expressing entropy in terms of the given variables and using the chain rule to simplify the differentiation. Comparing the resulting expression with the answer choices, we find that option (3) is the correct representation of the temperature.