How many times should the volume of 5 mol be increased?

How many times must the volume of 5 moles of an ideal gas increase during isothermal expansion if its entropy increases by 57.6 J/K?

Problem 20323. Detailed solution with a brief record of the conditions, formulas and laws used in the solution, derivation of the calculation formula and answer. If you have any questions regarding the solution, please write. I try to help.

To solve this problem, it is necessary to use the equation of state of an ideal gas, as well as the law of conservation of energy and the formula for changing entropy.

The condition of the problem states: it is necessary to find how many times the volume of 5 moles of an ideal gas must be increased during isothermal expansion if its entropy increases by 57.6 J/K.

In this case, since the process occurs during isothermal expansion, the gas temperature will remain unchanged. Therefore, we can use the ideal gas equation of state to find the volume of the gas in the initial and final states.

For the initial state we have: V1 = nRT/P, where n = 5 mol, R is the universal gas constant, T is temperature, P is pressure.

For the final state we have: V2 = nRT/(P+ΔP), where ΔP is the change in pressure during isothermal expansion.

The law of conservation of energy for an isothermal process has the form: Q = W, where Q is the thermal action and W is the work done by the gas.

From the formula for the change in entropy, we can express the change in thermal action: ΔQ = TΔS.

Thus, we can express the change in the work of a gas through a change in thermal action: W = -ΔQ = -TΔS.

Substituting the obtained expressions for the gas work and gas volumes in the initial and final states into the energy conservation equation, we obtain: -TΔS = PΔV, where ΔV = V2 - V1.

Based on the formula for the change in volume during an isothermal process (P1V1 = P2V2), we can express ΔP in terms of P1 and P2: ΔP = P1 - P2 = P1 - P1V1/V2.

Substituting the resulting expression for ΔP into the energy conservation equation, we obtain: -TΔS = P1(V2 - V1)/V2 + P1.

Expressing V2 in terms of V1 and the volume expansion factor k = V2/V1, we obtain: k = 1/(1 - ΔP/P1) = 1 + ΔV/V1.

So, we have obtained the formula for the coefficient of increase in the volume of an ideal gas during isothermal expansion: k = 1 + (TΔS)/(P1V1).

By substituting the known values ​​(T, ΔS, P1, V1) into this formula, you can find the desired volume increase factor.

Thus, the answer to the problem will depend on the values ​​of temperature, pressure and initial volume, which are not indicated in the condition. If you provide these values, I can help you solve the problem.

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In order to answer the question of how many times it is necessary to increase the volume of 5 moles, you need to know what substance this number of moles belongs to. A mole is a unit of measurement for the amount of a substance, so to answer the question you need to know the molar mass of the substance that is contained in 5 moles.

Without this information, it is impossible to determine exactly how many times the volume needs to be increased. If we assume that we know the molar mass of a substance, then to determine the required increase in volume it is necessary to know its density. After this you can use the formula:

V2 = (m / p) * k,

where V2 is the required volume, m is the mass of the substance, p is the density of the substance, k is the coefficient of increase in volume.

Thus, to answer the question it is necessary to know the molar mass and density of the substance, as well as the coefficient of increase in volume. Without this information, it is impossible to determine how many times the volume of 5 moles needs to be increased.

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