A balloon originally has a volume of 4.39L
at 44 C and a pressure of 729 torr. What temperature must the balloon be cooled to to reduce its volume to 3.78L at the same pressure?
(A) 38 C
(B) 72.9 C
(C) 0 C
(D) 546 C
(E) 273 C
Is the correct answer (E) 273 C?
To find the temperature at which the balloon needs to be cooled to reduce its volume to 3.78L at the same pressure, we can use the combined gas law equation:
(P1 x V1) / T1 = (P2 x V2) / T2
Given:
P1 = 729 torr
V1 = 4.39L
T1 = 44°C + 273 (convert to Kelvin)
P2 = 729 torr
V2 = 3.78L
T2 = ?
First, let's convert the initial temperature to Kelvin: T1 = 44°C + 273 = 317 K
Now, we can rearrange the equation to solve for T2:
T2 = (P2 x V2 x T1) / (P1 x V1)
Plugging in the values:
T2 = (729 torr x 3.78 L x 317 K) / (729 torr x 4.39 L)
Simplifying the equation:
T2 = (229816.46) / 3209.31
T2 ≈ 71.66 K
Converting back to Celsius:
T2 ≈ 71.66 K - 273 ≈ -201.34°C
The correct answer is not (E) 273°C. The correct answer is (C) 0°C.
To find the temperature at which the balloon must be cooled to reduce its volume, we can use the ideal gas law equation: 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 calculate the initial number of moles using the initial volume, pressure, and temperature. We can rearrange the ideal gas law equation to solve for n:
n = PV / RT
Given:
Initial volume (V1) = 4.39 L
Pressure (P) = 729 torr
Temperature (T) = 44 °C = 44 + 273.15 K (convert temperature to Kelvin)
Ideal gas constant (R) = 0.0821 L·atm/(mol·K) (using atm instead of torr for consistency)
Calculating the initial number of moles (n1):
n1 = (P * V1) / (R * T)
Next, let's rearrange the equation to solve for the final temperature (T2) using the final volume (V2) and the same pressure (P):
T2 = (P * V2) / (n1 * R)
Given:
Final volume (V2) = 3.78 L
Pressure (P) = 729 torr
Number of moles (n1) = calculated in the previous step
Ideal gas constant (R) = 0.0821 L·atm/(mol·K)
Now let's plug in the values and calculate T2:
T2 = (729 * 3.78) / (n1 * 0.0821)
To determine the correct answer choice, we need to calculate n1 and substitute it into the equation for T2.
n1 = (729 * 4.39) / (0.0821 * (44 + 273.15))
= 0.2507 mol
Now plug in the value of n1 into the equation for T2:
T2 = (729 * 3.78) / (0.2507 * 0.0821)
= 13796.26
Therefore, the temperature must be cooled to approximately 13796.26 K.
However, we need to convert this temperature to Celsius:
T2 (°C) = 13796.26 K - 273.15
= 13523.11 °C
Since none of the answer choices match exactly, we can check which one is the closest. The closest answer choice is (D) 546 °C.
So, based on the calculations, the correct answer is (D) 546 °C.