Chapter 1, Problem 1C.3E

Interpretation Introduction

Interpretation:

Pressure exerted by $1.00molC{H}_4$ has to be calculated using the value of virial equation of state and the given data.

Concept Introduction:

Ideal gas Equation:

Any gas is described by using four terms namely pressure, volume, temperature and the amount of gas. Thus combining three laws namely Boyle’s, Charles’s Law and Avogadro’s Hypothesis the following equation could be obtained. It is referred as ideal gas equation.

  $\begin{array}{l}\text{V }\propto \frac{\text{nT}}{\text{P}}\\ \text{V = R}\frac{\text{nT}}{\text{P}}\\ \text{PV = nRT}\end{array}$       $\begin{array}{l}\text{where,}\\ \text{n = moles}\text{of}\text{gas}\\ \text{P = pressure}\\ \text{T = temperature}\\ \text{R = gas }\text{constant}\text{and }\text{V=}\text{Volume}\end{array}$

Virial equation: This is an equation of state for real gases in Z as a power series in terms of $\frac{1}{V_m}$ for a pure gas.

  $Z=\frac{P{V}_m}{RT}=1+\frac{B}{V_m}+\frac{C}{V_m^2}+\mathrm{.....}$

B and C are the second and third virial coefficients and both are temperature dependent.

Van der Waals equation:

The deviation from ideal gas law was accounted using two constants, a and b and was given by an equation called van der Waals equation which is similar to ideal gas law as follows,

   $\left(\text{P+}\frac{{\text{n}}^{\text{2}}\text{a}}{{\text{V}}^{\text{2}}}\right)\left(\text{v-nb}\right)\text{=}\text{nRT}$

  Where,

a and b are constants

P is the pressure

V is the volume

T is the temperature

R is molar gas constant

n is the mole

Using van der Waal’s equation, the pressure can be determined as,

  $P=\text{}\frac{nRT}{V-nb}-a{\left(\frac{n}{V}\right)}^2$