2. A sample of a gas occupied a volume of 1.40 liters when the pressure was 768 torr and the temperature was 26.9 °C. The volume of the system was readjusted to 2.16 liters by changing the temperature. What was the temperature in the system?
To find the temperature in the system, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure (in atm)
V = volume (in liters)
n = number of moles
R = ideal gas constant (0.0821 L·atm/(mol·K))
T = temperature (in Kelvin)
First, let's convert the given pressure from torr to atm:
1 atm = 760 torr
So, 768 torr = 768/760 atm = 1.0118 atm.
Now, we can solve the equation for the initial conditions (before the volume change):
P1V1 = nRT1
Using the given values:
P1 = 1.0118 atm
V1 = 1.40 L
R = 0.0821 L·atm/(mol·K)
Solving the equation for T1, we get:
T1 = (P1V1) / (nR).
Since we don't have the number of moles (n), we need to find it using the ideal gas law equation again:
P1V1 = nRT1
Solving for n:
n = (P1V1) / (RT1).
Since the gas sample is the same before and after the change in volume (P, n, and R remain constant), we can use the number of moles we just calculated to find the temperature after the volume change (T2):
P2V2 = nRT2
Solving for T2:
T2 = (P2V2) / (nR).
Now, let's plug in the values we have:
P2 = P1 = 1.0118 atm (since the pressure remains constant)
V2 = 2.16 L (as given in the question)
So, the equation becomes:
T2 = (1.0118 atm * 2.16 L) / (n * 0.0821 L·atm/(mol·K)).
Plug in the value of "n" we calculated earlier to find the final temperature (T2) in the system.