from the first law of thympdynsmics, show that in the isothermal expansion of comprrssion of an ideal gas, the heat energy entering the system is equal to the work done by the gas and that these are RTln(V2- V1)

To show that in the isothermal expansion or compression of an ideal gas, the heat energy entering the system is equal to the work done by the gas, we can use the first law of thermodynamics.

The first law of thermodynamics states that the change in internal energy (ΔU) of a system is equal to the heat energy (Q) added to the system minus the work (W) done by the system:

ΔU = Q - W

For an isothermal process, the temperature (T) of the gas remains constant. Therefore, the change in internal energy (ΔU) is zero since the internal energy of an ideal gas is solely dependent on its temperature:

ΔU = 0

Substituting this into the first law equation, we get:

0 = Q - W

If we rearrange the equation, we can solve for work done (W):

W = Q

This equation tells us that in an isothermal process, the work done by the gas is equal to the heat energy entering the system.

Now, to derive the expression RTln(V2 - V1), we can use the ideal gas law, which states:

PV = nRT

Where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.

In an isothermal process, the temperature remains constant, so the ideal gas law can be rearranged as:

P1V1 = P2V2

Where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.

Now, let's consider the work done by the gas during compression or expansion. The work done by a gas is given by the equation:

W = ∫PdV

In an isothermal process, we can substitute the ideal gas law equation, P1V1 = P2V2, into the work equation:

W = ∫(P1V1/P1V)dV

W = ∫ V1/V dV

Integrating this equation gives us:

W = V1 ln(V2/V1)

Finally, we can substitute W = Q (from the earlier derivation) to find:

Q = V1 ln(V2/V1)

Since Q represents the heat energy entering the system, we can use it to calculate the value by multiplying by the gas constant R and the temperature T:

Q = nRT ln(V2/V1)

In this case, n is the number of moles of gas. Dividing both sides by n, we can simplify the equation as:

Q = RT ln(V2/V1)

Therefore, the expression for the heat energy entering the system in an isothermal expansion or compression of an ideal gas is equal to RT ln(V2 - V1).