Question

Use the van der Waals equation and the ideal gas equation to calculate the volume of 1.000 mol of neon at a pressure of 500.0 atm and a temperature of 355.0 K. Explain why the two values are different. (Hint: One way to solve the van der Waals equation for V is to use successive approximations. Use the ideal gas law to get a preliminary estimate for V.)

Answer 1

`v^(VdW)=0.0704(L)/(mol)`

`v^(ideal)=0.05816(L)/(mol)`

Differences:

- The gas is at a very high pressure, which causes strong nonidealities.

- Attraction and repulsion actually phenomena exist among molecules, which is not considered by the ideal gas equation.

Step-by-step explanation:

Hello,

In this case, the VdW equation takes the typical form:

`p=(RT)/(v-b)-(a)/(v^2)`

Whereas
`a` and
`b` are referred to the attraction and repulsion effect, in such a way, for neon, such values are 0.208 bar*L²/mol² and 0.01672 L/mol respectively. However, with the given information, the polynomic form of the VdW equation is:

`v^3-((RT)/(P)+b)v^2+(a)/(p) v-(ab)/(p) =0`

Which in terms of the molar volume becomes:

`v^3-((0.083(bar*L)/(mol*K)*355.0K)/(506.625bar)+0.01672(L)/(mol) )v^2+((0.208(bar*L^2)/(mol^2) )/(506.625bar))v-(0.208(bar*L^2)/(mol^2)*0.01672(L)/(mol))/(506.625bar) =0`

`v^3-0.07488v^2+0.0004106v-0.000006865=0`

Thus, by solving via solver the roots, two are imaginary and the feasible molar volume is:

`v^(VdW)=0.0704(L)/(mol)`

Now, the volume obtained from the ideal gas equation is:

`v^(ideal)=(RT)/(p)=(0.083(bar*L)/(mol*K)*355.0K)/(506.625bar) =0.05816(L)/(mol)`

In this case, the two values are different because:

- The gas is at a very high pressure, which causes strong nonidealities.

- Attraction and repulsion actually phenomena exist among molecules, which is not considered by the ideal gas equation.

Best regards.

Answer 2

Ideal gas: V = 0.05826L

Van der Waals: V = 0.071L

Step-by-step explanation:

Given Data:

Number of Mole = 1.000

Pressure = 500atm

temperature = 355.0k

R = 0.082057L atm mol^-1K^-1

1. Using the ideal gas equation, the volume can be calculated as;

PV = nRT

V = nRT/P

= (1 * 0.082057 * 355)/500

= 29.130/500

V = 0.05826L

2. Using the van der Waals equation to calculate the volume; we have

[P + (n²a/V²)](V-nb) = nRT

For neon gas,

a = 0.2135

b = 0.01709

Substituting, we have

[500 + (1² * 0.2135/V²)] (V-1 * 0.01709) = 1 * 0.082057 * 355

[ 500 + (0.2135/V²)] ( V- 0.01709) = 29.130

V = 0.071L

The ideal gas molecules do not interact with each other but the real gas molecules interact. Therefore, the difference in volume is due to interaction between atoms of real gas but not in the ideal gas.