Chapter U3.17, Problem 7E

Interpretation Introduction

Interpretation: Three methods need to be explained to get the pressure of gas 0.50 atm, if 1 mol of gas has volume 22.4 L at STP.

Concept Introduction: The ideal gas equation is represented as follows:

  $PV=nRT$

Here, P is pressure, V is volume, n is number of moles and T is temperature of the gas.

R is Universal gas constant with value $0.082\text{ L atm/K mol}$ .

Explanation of Solution

At STP, the temperature and pressure values are 273.15 K and 1 atm respectively. The given number of moles and volume of the gas is 1 mol and 22.4 L respectively.

Now, final pressure is given 0.5 atm which is half of the initial pressure.

The ideal gas equation is represented as follows:

  $PV=nRT$

Here, R is the only constant thus, it cannot be changed.

According to above equation, if V and T are kept constant the pressure will be directly proportional to the number of moles. Thus, to make final pressure half of the initial value, the number of moles also needs to be divided by 2.

Calculate pressure keeping other variables same, but number of moles equals to half of the initial value that is 0.5 mol.

  $P=\frac{nRT}{V}=\frac{\left(0.5\text{ mol}\right)\left(0.082\text{ L atm/K mol}\right)\left(273.15\text{ K}\right)}{\left(22.4\text{ L}\right)}=0.5\text{ atm}$

Similarly, if n and T are kept constant the pressure of the gas is inversely proportional the volume. Thus, to make the final pressure half the initial value, volume of the gas should be doubled.

Calculate pressure keeping other variables same, but volume of the gas equals to twice the initial value that is 44.8 L.

  $P=\frac{nRT}{V}=\frac{\left(1\text{ mol}\right)\left(0.082\text{ L atm/K mol}\right)\left(273.15\text{ K}\right)}{\left(44.8\text{ L}\right)}=0.5\text{ atm}$

The third way is to change the temperature. Since, P and T are directly proportional, if temperature decreases by a factor of 2 and other variables remain constant thus, P also becomes half.

  $P=\frac{nRT}{V}=\frac{\left(1\text{ mol}\right)\left(0.082\text{ L atm/K mol}\right)\left(136.5\text{ K}\right)}{\left(22.4\text{ L}\right)}=0.5\text{ atm}$

Conclusion

Therefore, the three ways to make pressure equals to 0.5 atm are: decreasing number of moles by a factor of 2, decreasing temperature by a factor of 2 and increasing volume two times.