# Density of Gases with Examples

**Density of Gases with Examples**

Density of gases is too small with respect to solid and gas phases. We can find density with following formula;

**d(gas)=m(gas)/V(gas)**

if we substitute it into the ideal gas law;

P.V=n.R.T where n=mass/molar mass

P.V=(m/Mm).R.T

P.Mm=(m/V).R.T

P.Mm=d.R.T

**d=(P.Mm)/(R.T)**

As you can see from the formula; density of gases is directly proportional to pressure and molar mass and inversely proportional to temperature.

**Example:** Find density of C_{4}H_{8} at 273 ^{0}C and under 2 atm pressure. (H=1, C=12)

**Solution:** we make unit conventions first;

T=273+273=546 K

P=2 atm

C_{4}H_{8}=4.12+8.1=56 g/mol

Using formula given above;

P.Mm=d.R.T

2.56=d.(22,4/273).546

d=2,5 g/L

**Example:** If we add some CH_{4} to container given below under constant temperature; which ones of the following statements are true related to gases in this container? (He=4, C=12, H=1)

**I.** Density of mixture increases

**II.** Volume increases

**III.** Pressure increases

**Solution:**

Molar mass of CH_{4}=12+4.1=16

Since piston of container is dynamic, when we add CH_{4}, volume of mixture increases. Molar mass of CH_{4} is greater than He, thus density of mixture also increases.

P.Mm=d.R.T

Increasing in the volume of gas balance pressure and it stays constant.

**Example:** Which ones of the graphs are true for ideal gas.

**Solution:**

**I.** Using ideal gas law;

P.V=n.R.T

P=n.R.T/V

Since R, n and V are constant, P is directly proportional to temperature. Graph I is true.

**II.** Molar volume is V/n. Using ideal gas law;

V/n=R.T/P

Since R, P and T are constant V/n must be constant. Thus second graph is false, line showing relation between V and n must be parallel to n.

**III.** We write ideal gas law for density;

d=P.Mm/R.T

Mm, R and P are constant , thus d is inversely proportional to T. III. graph is true.

**Example:** Graph given below shows density vs. volume relation of X(gas) at 0^{0}C. If the pressure of X(gas) at point A is 1 atm, which ones of the following statement are true for this gas.

**I.** n=1mol

**II.** Pressure at point B is 0,5 atm

**III.** Molar mass of gas is 56 g

**Solution:**

**I.** Ideal gas law at point A;

P_{A}.V_{A}=n.R.T

T=0 ^{0}C or 273 K

V=11,2 Liters

P=1 atm

n=P_{A}.V_{A}/R.T=(1.11,2)/(22,4/273).273)=0,5mol

I is false.

**II.** n and T are constant, thus we can write;

P_{A}.V_{A}=P_{B}.V_{B}

(1.11,2)=P_{B}.22,4

P_{B}=0,5 atm, II is true

**III.** density at point A is;

d_{A}=P_{A}.Mm/R.T

Mm=(dA.R.T)/PA=(2,5.(22,4/273).273)/1

Mm=56 g/mol

1 mol gas contains 56 g, so III is true.

**Example:** Which ones of the following statements are true for He and O_{2} gases under same temperature. (He=4, O=16)

**I.** n_{He}=4n_{O2}

**II.** d_{He}=8d_{O2}

**III.** Average kinetic energies of He and O_{2} are equal.

**Solution:**

We find pressures of gases using manometers.

P_{He}=2h+2h=4h

P_{O2}=2h

To find relation between number of moles of gases we use ideal gas law.

P_{He}.V_{He}=n_{He}.R.T_{He}

n_{He}=4h.2V/R.T

P_{O2}.V_{O2}_{}=n_{O2}.R.T_{O2}

n_{O2}=2h.V/R.T

Ratio of nHe and n_{O2}

**n _{He}/n_{O2}=4/1** Thus, I is true.

We find density of gases again using ideal gas law.

Molar mass of He=4 and Molar mass of O_{2}=2.16=32

d_{He}=P_{He}.M_{He}/R.T

d_{He}=4h.4/R.T

d_{O2}=P_{O2}.M_{O2}/R.T

d_{O2}=2h.32/R.T

Ratio de las densidades;

**d _{He}/d/O_{2}=1/4** so, II is false.

**III.** Since temperature of gases same, their average kinetic energies are also same. III is true.