To solve this problem, we can use the first law of thermodynamics for an adiabatic process:

ΔU = q + w

Where:
ΔU is the change in internal energy of the gas
q is the heat transferred to the gas
w is the work done by the gas

Since the process is adiabatic, q = 0, so the equation becomes:

ΔU = w

The work done by the gas can be calculated using the equation:

w = -PΔV

Where:
P is the pressure
ΔV is the change in volume

Given that the gas is expanded adiabatically at constant pressure until it doubles its volume, we can calculate the change in volume:

ΔV = V2 - V1 = 2V1 - V1 = V1

Now, we can substitute the values into the equation for work:

w = -PΔV = -P(V1)

The change in internal energy can be calculated using the equation:

ΔU = ncΔT

Where:
n is the number of moles of the gas
c is the molar heat capacity at constant volume
ΔT is the change in temperature

Given that the gas is an ideal gas, we can use the ideal gas law to calculate the number of moles:

PV = nRT

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

Rearranging the equation, we can solve for n:

n = PV / RT

Substituting the given values, we can calculate the number of moles.

Now, we can substitute the values into the equation for the change in internal energy:

ΔU = ncΔT = (PV / RT)cΔT

Since the process is adiabatic, ΔU = w. Therefore, we have:

(PV / RT)cΔT = -P(V1)

Simplifying the equation, we can solve for ΔT:

ΔT = -RT / cV1

Substituting the given values, we can calculate the change in temperature. Adding this change to the initial temperature, we can find T2.

Let's calculate the values step by step:

Given:
cv = 5/2R
P1 = 5 atm
V1 = V
T1 = 25°C + 273.15 = 298.15 K
V2 = 2V

Step 1: Calculate the number of moles (n)
Using the ideal gas law:
n = PV / RT
n = (P1V1) / (RT1)
n = (5 atm)(V) / (0.0821 L.atm/mol.K)(298.15 K)
n = 0.167V

Step 2: Calculate the change in temperature (ΔT)
Using the equation:
ΔT = -RT / cV1
ΔT = -(0.0821 L.atm/mol.K)(298.15 K) / ((5/2)(0.167V))
ΔT = -0.984 V

Step 3: Calculate T2
T2 = T1 + ΔT
T2 = 298.15 K - 0.984 V

Since