An ideal gas with r(Gama) = 1.5 occupies a volume of 2.0 L. The gas expands isothermally to a volume of 3.0 L. Alternatively, the gas reaches the same final state by first adiabatically expanding to the final volume then heating isochorically to the final temperature. For which path is the heat transferred to the gas greatest and by what percent of the lesser amount is it greater?

Question

An ideal gas with r(Gama) = 1.5 occupies a volume of 2.0 L. The gas expands isothermally to a volume of 3.0 L. Alternatively, the gas reaches the same final state by first adiabatically expanding to the final volume then heating isochorically to the final temperature. For which path is the heat transferred to the gas greatest and by what percent of the lesser amount is it greater?

Answer 1

To determine which path has the greatest heat transferred to the gas and by what percent it is greater, we need to compare the heat transferred for both the isothermal and adiabatic paths.

Let's begin by calculating the heat transferred in the isothermal path.

In an isothermal process, the temperature remains constant. We can use the ideal gas law to calculate the initial and final pressures of the gas.

For the initial state:
PV = nRT

Given:
Volume (V1) = 2.0 L
R (Universal gas constant) = 8.314 J/(mol·K)
Temperature (T1) = constant
Number of moles (n) = constant

From the ideal gas law, we can rearrange the equation to solve for the initial pressure (P1) as:
P1 = (nRT1) / V1

Similarly, for the final state:
Volume (V2) = 3.0 L

Using the same equation, we can solve for the final pressure (P2) as:
P2 = (nRT1) / V2

Now, let's calculate the work done in the isothermal process using the formula:
W = nRT1 * ln(V2/V1)

Since the process is isothermal, the heat transferred (Q) is equal to the work done:
Q(isothermal) = W = nRT1 * ln(V2/V1)

Next, let's calculate the heat transferred in the adiabatic path.

In an adiabatic process, no heat is transferred to or from the gas. Therefore, the equation for heat transferred is simply:
Q(adiabatic) = 0

Now, let's compare the two heat transfers.

The heat transferred in the isothermal path is given by:
Q(isothermal) = nRT1 * ln(V2/V1)

The heat transferred in the adiabatic path is:
Q(adiabatic) = 0

Since Q(isothermal) is greater than Q(adiabatic) in this case, the heat transferred to the gas is greatest for the isothermal path.

To calculate the percentage by which it is greater, we can use the following formula:
Percentage increase = (Q(isothermal) - Q(adiabatic)) / Q(adiabatic) * 100

Substituting the values, we get:
Percentage increase = (Q(isothermal) - 0) / 0 * 100 = Infinity

Therefore, the heat transferred in the isothermal path is infinitely greater than that in the adiabatic path.