hot-air balloon is filled with air at a volume of 3.64 x 10^3 m^3 at 745 torr at 39C. the air in balloon is heated to 60C, causing balloon to expand to a volume of 4.50x 10^3 m^3. what is the ratio of the number of moles of the air in heated balloon to original number of moles of air in balloon? (hint: openings in balloon allow air to flow in and out, thus pressure in balloon is always same as atmosphere.
I would use PV = nRT
Use any number for P since it is constant. I would choose 1 atmosphere. I would change m^3 to liters although that probably isn't necessary since the problem asks for a ratio. Use R = 0.08205, convert T to Kelvin (K = C + 273). Calculate n for each set of parameters and take the ratio of the two values.
To solve this problem, we need to use the ideal gas law equation: PV = nRT.
Given:
Initial volume (V1) = 3.64 x 10^3 m^3
Initial temperature (T1) = 39°C = 312 K
Final volume (V2) = 4.50 x 10^3 m^3
Final temperature (T2) = 60°C = 333 K
Since the pressure inside the balloon is the same as the atmosphere, we can assume the pressure (P) is constant.
Now, let's calculate the number of moles of air using the ideal gas law equation for both initial and final conditions.
For initial conditions:
P1V1 = n1RT1
n1 = P1V1 / (RT1)
For final conditions:
P2V2 = n2RT2
n2 = P2V2 / (RT2)
Since the pressures are the same, we have:
n1 = n2
Now we can calculate the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon.
Ratio = n2 / n1
Ratio = (P2V2 / RT2) / (P1V1 / RT1)
Ratio = (P2V2 / P1V1) * (RT1 / RT2)
Plugging in the given values:
Ratio = (P2V2 / P1V1) * (T1 / T2)
Ratio = (745 torr * 4.50 x 10^3 m^3) / (745 torr * 3.64 x 10^3 m^3) * (312 K / 333 K)
Let's calculate this:
Ratio ≈ (3352500 torr * m^3) / (2711800 torr * m^3) * (312 K / 333 K)
Ratio ≈ 1.235
Therefore, the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon is approximately 1.235.
To solve this problem, we need to 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 in Kelvin.
First, let's convert the temperatures from Celsius to Kelvin:
Initial temperature (T1) = 39 + 273 = 312 K
Final temperature (T2) = 60 + 273 = 333 K
According to the problem, the pressure inside the balloon is the same as the atmospheric pressure, which is 745 torr.
To find the number of moles (n) in the original balloon, we need to rearrange the ideal gas law equation to solve for n:
n1 = (P1 * V1) / (R * T1)
Using the values given:
P1 = 745 torr
V1 = 3.64 x 10^3 m^3
R = 0.0821 L·atm/(mol·K) (I will assume atmospheric pressure is in atm, as torr is commonly converted to atm)
T1 = 312 K
Plugging in these values into the equation, we have:
n1 = (745 atm * 3.64 x 10^3 m^3) / (0.0821 L·atm/(mol·K) * 312 K)
Simplifying this gives us the number of moles in the original balloon.
Similarly, we can find the number of moles (n2) in the heated balloon using the final conditions:
n2 = (P1 * V2) / (R * T2)
Plugging in the values:
P1 = 745 torr
V2 = 4.50 x 10^3 m^3
R = 0.0821 L·atm/(mol·K)
T2 = 333 K
Simplifying this equation gives us the number of moles in the heated balloon.
Finally, to calculate the ratio of the number of moles of air in the heated balloon to the original number of moles, we divide n2 by n1:
Ratio = n2 / n1
Plug in the calculated values of n2 and n1 to find the ratio.