The balloon contains 0.30 mol of helium starts at a pressure of 125 kpa and rises to an altitude where the pressure is 55.0 kpa, maintaining a constant 300 {\rm K} temperature.
A)By what factor does its volume increase?
B)How much work does the gas in the balloon do?
p1•V1/T1 = p2•V2/T2á
T1 =T2á
V2/V1=p1/p2=125/55 =2.27þ
W =ν•R•T•ln(V2/V1) = =0.3•8.31•300•ln2.27=614 J.
A) To find the factor by which the volume increases, we can use the combined gas law:
P1V1 / T1 = P2V2 / T2
Where:
P1 = initial pressure = 125 kPa
V1 = initial volume (unknown)
T1 = initial temperature = 300 K
P2 = final pressure = 55.0 kPa
V2 = final volume (unknown)
T2 = final temperature = 300 K
Since the temperature remains constant, we can cancel out T1 and T2 from the equation:
P1V1 = P2V2
Rearranging the equation to solve for the volume ratio:
V2/V1 = P1/P2
Substituting the given values:
V2/V1 = (125 kPa) / (55.0 kPa)
V2/V1 = 2.273
Therefore, the volume of the balloon increases by a factor of approximately 2.273.
B) The work done by the gas can be calculated using the formula:
Work = PΔV
Where:
P = average pressure = (P1 + P2) / 2
ΔV = change in volume = V2 - V1
Substituting the given values:
P = (125 kPa + 55.0 kPa) / 2 = 90.0 kPa
ΔV = V2 - V1 = (2.273V1 - V1) = 1.273V1
Work = (90.0 kPa) * (1.273V1)
Note: The work is given in units of kilopascals * volume.
Please provide the value of the initial volume (V1) of the balloon to calculate the exact work done by the gas.
To answer the given questions, we can use the ideal gas law equation, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.
First, let's determine the initial volume of the balloon (V1). We can rearrange the ideal gas law equation to solve for V1:
V1 = (nRT) / P1
Where:
n = 0.30 mol (given)
R = ideal gas constant = 8.314 J/(mol K)
T = 300 K (given)
P1 = 125 kPa (given)
Plugging in the values:
V1 = (0.30 mol * 8.314 J/(mol K) * 300 K) / 125 kPa
Next, let's determine the final volume of the balloon (V2). We can use the same equation, but with the new pressure value (P2 = 55.0 kPa):
V2 = (nRT) / P2
Plugging in the values:
V2 = (0.30 mol * 8.314 J/(mol K) * 300 K) / 55.0 kPa
Now, let's calculate the increase in volume:
Volume Increase = V2 / V1
Finally, to determine the work done by the gas in the balloon, we can use the formula:
Work (W) = -P2(V2 - V1)
Where:
P2 = 55.0 kPa (final pressure)
V2 = final volume
V1 = initial volume
Plugging in the values, we can calculate the work done by the gas.