Question

Two moles of a monatomic ideal gas such as helium are compressed adiabatically and reversibly from a state (3 atm, 5 L) to a state with a pressure of 4 atm.

(a) Find the volume and temperature of the final state.

(a) V = 3.75 L, T = 315 K
(b) V = 5.00 L, T = 300 K
(c) V = 4.00 L, T = 300 K
(d) V = 3.00 L, T = 288 K

Answer 1

The volume of the final state is approximately 3.75 L, and the temperature is approximately 315 K.

Step-by-step explanation:

To find the volume and temperature of the final state, we can use the adiabatic process equations for an ideal gas. In an adiabatic process, the relation between pressure (P), volume (V), and temperature (T) can be described by the equation PV^y = constant, where y is the ratio of specific heats for a monatomic ideal gas (y = 5/3).

Using this equation, we can solve for the final volume (V₂) in terms of the initial volume (V₁) and initial pressure (P₁):

V₂ = (P₁/P₂)^(1/y) * V₁ = (3/4)^(1/5/3) * 5 L ≈ 3.75 L

Next, we can use the ideal gas law to find the final temperature (T₂) by rearranging the equation PV = nRT and solving for T:

T₂ = (P₂/P₁) * (V₁/V₂) * T₁ ≈ (4/3) * (5/3.75) * 300 K ≈ 315 K