# It turns out that the van der Waals constant b equals four times the total volume actually occupied by the molecules of a mole of gas. Using this figure, calculate the fraction of volume in a container actually ocupied by Ar atoms (a) at STP, (b) at 100 atm pressure and 0 degrees celsius. (Assume for simplicity that the ideal-gas equation still holds.)

thanks for any help =]

a gas

## To calculate the fraction of volume actually occupied by Ar atoms, we need to use the van der Waals equation and the given van der Waals constant, b. The van der Waals equation is:

(P + a(n/V)^2)(V - nb) = nRT

Where:

P = pressure

V = volume

n = number of moles

R = gas constant

T = temperature in Kelvin

a = van der Waals constant accounting for attractive forces

To calculate the fraction of volume, we need to determine the actual volume occupied by Ar atoms and divide it by the total container volume.

At STP (Standard Temperature and Pressure), we have:

P = 1 atm

T = 273.15 K

R = 0.0821 L atm/(mol K)

We can assume that n = 1 mol

Plugging these values into the van der Waals equation, we can solve for V:

(1 + a/(V^2))(V - 4V) = RT

Simplifying, we have:

(1 + a/V^2)(V - 4V) = RT

Now, we can solve for V. Rearranging the equation, we get:

(1 + a/V^2)(V - 4V) = RT

V - 4V + a/V - 4a/V^2 = RT

V(1 - 4 + a/V - 4a/V^2) = RT

V(-3 + a/V - 4a/V^2) = RT

V(-3V + a - 4a/V) = RT

-3V^2 + aV - 4a = RTV

-3V^2 + aV - RTV + 4a = 0

This is a nonlinear equation that requires numerical methods to solve. Once we find the value of V, we can calculate the fraction of volume actually occupied by Ar atoms by dividing the actual volume by the total container volume.

To calculate at 100 atm pressure and 0 degrees Celsius, we can use the same approach, but with different values of pressure and temperature.

P = 100 atm

T = 273.15 K (0 degrees Celsius is 273.15 Kelvin)

Again, we assume n = 1 mol and solve for V using the van der Waals equation. Once we find the value of V, we can calculate the fraction of volume actually occupied by the Ar atoms by dividing the actual volume by the total container volume.

Keep in mind that the van der Waals equation is an approximation and assumes ideal gas behavior with corrections for attractive forces (a) and excluded volume (b). It is worth noting that at high pressures and low temperatures, the deviation from ideal gas behavior becomes significant.