40. If 12.0 gram of O2 is required to inflate a balloon to a certain size at 27 degrees C, what mass of O2 is required to inflate it to the same size and pressure at 81 deg C?
I would do this the long way.
PV = nRT. We can ignore P and R since they are constants; therefore
V = nT
V = (12.0/32)*300 = about 112 but you do it more accurately.
Then 112/(273+81) = n = about 0.31
and 0.31 x 32 = about 10 g.
The short way is
(12/x) = (354/300)
x = 12 x (300/354) = ?
x =
Oh, boy! It looks like that balloon is in for quite the temperature change! So, let’s figure out how much O2 is needed.
We know that the amount of gas needed will be directly proportional to the temperature. Since the temperature has increased from 27°C to 81°C, we can use the following formula:
(V1/T1) = (V2/T2)
Here, V1 and T1 represent the initial volume and temperature, and V2 and T2 represent the final volume and temperature.
But wait, we don't need to worry about volume; we only need to find the mass of O2 required. So, let's change the formula up a bit:
(m1/T1) = (m2/T2)
Where m1 and T1 represent the initial mass and temperature, and m2 and T2 represent the final mass and temperature.
Plugging in the values we know:
(12.0g / 27°C) = (m2 / 81°C)
Now, let's solve for m2, shall we?
Cross-multiplying:
12.0g * 81°C = 27°C * m2
Doing some calculations:
972g°C = 27°C * m2
Finally, isolating m2:
m2 = 972g°C / 27°C
Drumroll, please! *ba-dum-tss*
After canceling out the units:
m2 ≈ 36g
So, to inflate the balloon to the same size and pressure at 81°C, you'll need approximately 36 grams of O2. Just keep that temperature in check, or your balloon might turn into a hot air balloon!
To calculate the mass of O2 required to inflate the balloon to the same size and pressure at 81 degrees Celsius, we can use the ideal gas law equation:
PV = nRT
Where:
P is the pressure of the gas
V is the volume of the gas
n is the number of moles of gas
R is the ideal gas constant
T is the temperature of the gas in Kelvin
First, let's convert the given temperatures to Kelvin:
27 degrees Celsius + 273.15 = 300.15 K
81 degrees Celsius + 273.15 = 354.15 K
Since the volume and pressure of the balloon are the same, we can ignore those variables. So the equation becomes:
n1/T1 = n2/T2
Now let's calculate the number of moles of O2 gas required to inflate the balloon at 27 degrees Celsius:
n1/300.15 K = 12.0 g / molar mass of O2
The molar mass of O2 is 32.00 g/mol.
Solving for n1:
n1 = (12.0 g / 32.00 g/mol) * 300.15 K
n1 = 112.54 mol
Now we can use the number of moles, n1, to calculate the mass of O2 required at 81 degrees Celsius:
n2/354.15 K = 112.54 mol / 300.15 K
Solving for n2:
n2 = (112.54 mol / 300.15 K) * 354.15 K
n2 = 132.49 mol
Finally, we can calculate the mass of O2 required at 81 degrees Celsius:
mass of O2 = n2 * molar mass of O2
mass of O2 = 132.49 mol * 32.00 g/mol
mass of O2 = 4239.68 g
Therefore, 4239.68 grams of O2 is required to inflate the balloon to the same size and pressure at 81 degrees Celsius.
To solve this problem, we can use the ideal gas law equation, which is:
PV = nRT
Where:
P = pressure,
V = volume,
n = number of moles,
R = ideal gas constant (0.0821 L·atm/mol·K),
T = temperature in Kelvin.
First, let's calculate the number of moles of O2 required to inflate the balloon at 27 degrees C. We can use the given mass of O2 (12.0 grams) and the molar mass of O2 (32.0 g/mol).
Number of moles (n) = Mass (m) / Molar mass (M)
n = 12.0 g / 32.0 g/mol
n = 0.375 mol
Now, we can calculate the volume of the balloon at 27 degrees C. Since the pressure and volume are constant, we can use the equation PV = nRT and solve for V.
V = nRT / P
We know that the temperature is 27 degrees C, which is 300 Kelvin (27 degrees C + 273.15).
V = (0.375 mol) * (0.0821 L·atm/mol·K) * (300 K) / P
V = 9.31 L / P
Now, we want to find the mass of O2 required to inflate the balloon to the same size and pressure at 81 degrees C. We can use the same equation, but this time the temperature is 81 degrees C, which is 354 Kelvin (81 degrees C + 273.15).
V = nRT / P (at 81 degrees C)
To inflate the balloon to the same size and pressure, the volume should be constant. Therefore, the volume (V) is the same as before (9.31 L).
Plugging in these values, we have:
9.31 L = (n) * (0.0821 L·atm/mol·K) * (354 K) / P
Now, we need to solve for the number of moles (n) at 81 degrees C. Rearranging the equation, we get:
n = (9.31 L) * P / (0.0821 L·atm/mol·K * 354 K)
Finally, we can calculate the mass of O2 required at 81 degrees C using the number of moles (n) and the molar mass (M).
Mass (m) = n * M
m = [(9.31 L) * P / (0.0821 L·atm/mol·K * 354 K)] * M
Since we don't know the pressure (P), we can't calculate the exact mass of O2 required without that information. However, if the pressure remains constant, we can use the equation to find the mass of O2 required at the given pressure and temperature.