Answer:
a. 0.00189 kg
b. 18.552 mN
c. 1.363 kN
d. Since the molecular density is high, the force exerted by the sample is thus high.
Step-by-step explanation:
(a) Find the mass of the gas. kg
Using PV = mRT/M where P = pressure on gas = atmospheric pressure = 1.013 × 10⁵ Pa, V = volume of gas = L³ where L = length of cube = 11.6 cm = 0.116 cm,m = mas of gas, R = molar gas constant = 8.314 J/mol-K, T = temperature of gas = 291 K and M = molar mass of gas = 28.9 g/mol
So, m = PVM/RT = PL³M/RT
Substituting the values of the variables into the equation, we have
m = PL³M/RT
= 1.013 × 10⁵ Pa × (0.116)³ × 28.9 g/mol/ 8.314 J/mol-K × 291 K
= 0.0457 × 10⁵ Pa g/mol/2419.374J/mol
= 1.89 × 10⁻⁵ × 10⁵ g
= 1.89 g
= 1.89 × 10⁻³kg
= 0.00189 kg
(b) Find the gravitational force exerted on it. mN
The gravitational force, F exerted on it is its weight W
So, F = W = mg where m = mass of gas = 1.89 × 10⁻³ kg and g = acceleration due to gravity = 9.8 m/s²
F = mg
= 1.89 × 10⁻³ kg × 9.8 m/s²
= 18.522 × 10⁻³ kgm/s²
= 18.552 × 10⁻³ N
= 18.552 mN
(c) Find the force it exerts on each face of the cube. kN
Since pressure, P = F/A where F = force exerted on each face and A = area of each face = L² where L = length of side of cube = 11.6 cm = 0.116 m
So, F = PA since P = atmospheric pressure = 1.013 × 10⁵ Pa,
F = PL²
= 1.013 × 10⁵ Pa (0.116 m)²
= 0.01363 × 10⁵ N
= 1.363 × 10³ N
= 1.363 kN
(d) Why does such a small sample exert such a great force
To answer this question, we need to find the density of the gas in the cube.
So density of gas,ρ = m/V where m = mass of gas = 1.89 g and V = volume of gas = L³ and L = length of side of cube = 11.6 cm
ρ = m/V = m/L³ = 1.89 g/(11.6 cm)³ = 1.89 g/1560.896 cm³ = 0.00121 g/cm³
We now find the number of moles of gas in a cm³ by dividing its density by its molar mass.
So n = ρ/M = 0.00121 g/cm³ ÷ 28.9 g/mol = 23687.67 mol/cm³
Since there are 6.022 × 10²³/mol, we find the number of molecules in a cm³ which is n × 6.022 × 10²³/mol = 23687.67 mol/cm³ × 6.022 × 10²³/mol
= 143731.1 × 10²³ molecules/cm³
= 1.437311 × 10²⁸ molecules/cm³
≅ 1.44 × 10²⁸ molecules/cm³
Since the molecular density is high, the force exerted by the sample is thus high.