Question

A hot-air balloon is filled with air to a volume of 4.00 3 103 m3 at 745 torr and 218C. The air in the balloon is then heated to 628C, causing the balloon to expand to a volume of 4.20 3 103 m3 . What is the ratio of the number of moles of air in the heated balloon to the original number of moles of air in the balloon

Answer 1

Answer : 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, 0.5735

Explanation :

First we have to calculate the original number of moles of  air in the balloon.

Using ideal gas equation:

PV = nRT

or,

$n_1=\frac{PV_1}{RT_1}$

where,

$n_1$ = original number of moles of air in the balloon = ?

P = pressure of gas = 745 torr = 99308.5 Pa     (conversion used : 1 torr = 133.3 Pa)

$V_1$ = volume of gas = $4.00\times 10^3m^3$

$T_1$ = temperature of gas = $218^oC=273+218=491K$

R = gas constant = $8.314Pa.m^3/mol.K$

Now put all the given values in the above equation, we get:

$n_1=\frac{(99308.5 Pa)\times (4.00\times 10^3m^3)}{(8.314Pa.m^3/mol.K)\times (491K)}$

$n_1=97309.4mol$

Now we have to calculate the number of moles of  air in heated balloon.

Using ideal gas equation:

PV = nRT

or,

$n_2=\frac{PV_2}{RT_2}$

where,

$n_2$ = number of moles of air in heated balloon = ?

P = pressure of gas = 745 torr = 99308.5 Pa     (conversion used : 1 torr = 133.3 Pa)

$V_2$ = volume of gas = $4.20\times 10^3m^3$

$T_2$ = temperature of gas = $626^oC=273+626=899K$

R = gas constant = $8.314Pa.m^3/mol.K$

Now put all the given values in the above equation, we get:

$n_2=\frac{(99308.5 Pa)\times (4.20\times 10^3m^3)}{(8.314Pa.m^3/mol.K)\times (899K)}$

$n_2=55804.1mol$

Now we have to 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.

$\frac{n_2}{n_1}=\frac{55804.1mol}{97309.4mol}=0.5735$

Thus, 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, 0.5735