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?

To determine which path results in the greatest heat transferred to the gas, we can compare the heat transferred for each path.

First, let's consider the isothermal expansion. For an isothermal process, the equation relating heat transfer (Q), temperature (T), and the ideal gas constant (R) is:

Q = nRT * ln(V2/V1),

where n is the number of moles of gas, R is the ideal gas constant, V1 is the initial volume, and V2 is the final volume.

Since the gas is ideal, we can use the ideal gas law to express nRT in terms of pressure (P) and volume (V):

nRT = PV.

Substituting this expression into the equation for Q, we get:

Q = P * V * ln(V2/V1).

Next, let's consider the second path, which involves adiabatic expansion followed by isochoric heating.

For an adiabatic process, the equation relating pressure (P), volume (V), and the heat capacity ratio (γ) of the gas is:

P * V^γ = constant.

In this case, γ is given as 1.5.

We know the initial volume is 2.0 L, and the final volume is 3.0 L. Let's call the volume at the end of adiabatic expansion V3.

Since the volume is constant during isochoric heating, the final volume of 3.0 L will remain unchanged. Let's call the final pressure at this volume P4.

Using the equation for adiabatic expansion, we can find P3:

P2 * V2^γ = P3 * V3^γ.

Since the final volume is 3.0 L, V3 is also 3.0 L.

Now, using the equation for isochoric heating, we can find P4:

P3 * V3^γ = P4 * V4^γ.

Since the volume remains constant, V4 is also 3.0 L.

Now, let's find the heat transferred during adiabatic expansion:

Q1 = P3 * V3 * (V3/V2)^(γ-1).

Finally, let's find the heat transferred during isochoric heating:

Q2 = P4 * V4 * ln(V2/V4).

Now that we have expressions for Q1 and Q2, we can compare them and determine which one is greater.

To find the percentage by which Q2 is greater than Q1, we can use the formula:

Percentage = ((Q2 - Q1) / Q1) * 100.

By evaluating the expressions for Q1 and Q2 and calculating the percentage, we can determine the path that results in the greatest heat transfer and by what percentage it is greater.