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# Diamonds Aren't Forever

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Age 16 to 18

Challenge Level

The ideal gas equation is a simple equation which can be used to model certain gases under certain conditions. In this question, assume that it is always applicable. The equation is as follows:

$$pV = nRT$$

where

$p$ = pressure

$V$ = volume

$n$ = number of moles

$R$ is 8.314 JK$^{-1}$mol$^{-1}$

$T$ = temperature

I vaporise a diamond using a laser, such that the gas fills 55000 cm$^3$ at a pressure of 900 mmHg and a temperature of 49$^\circ$C.

*How many moles of carbon are there in my original sample of diamond? Note that data for this question are at the bottom of the page.*

*Assuming that the diamond can be modelled as a sphere of density 3.52* $\times$ *10*$^{9}$ *mg/m*$^3$*, what would the radius of my diamond have been in cm?*

I now cool the vapour to -20$^\circ$C at a constant volume. There is no other heat transfer to or from the gas.

*What is the new pressure in Nm*$^{-2}$*?*

*If I allow the volume of the gas to double (against a vacuum), what would the new pressure be?*

*If the expansion had not been against a vacuum, would the new pressure be larger or smaller than that calculated previously? Why?*

The gas is now returned to its original temperature and volume, and 5 moles of air are introduced into the container.

*What is the new pressure of the gas in container?*

*Given that air is 0.93% argon (by volume), what is the partial pressure of argon?*

*If the argon molecules are evenly distributed in the container, what volume does each molecule occupy?*

I now try to compress the gas into the smallest volume possible.

*Making suitable modelling assumptions, estimate what this smallest volume would be*.

Data:

1 atm = 760 mmHg

1 bar = 10$^5$ Pa = 0.987 atm

N radius = 75 pm

C radius = 77 pm

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.