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Question
Using the equation of state pV = nRT; show that at a given temperature density of a gas is proportional to gas pressure p.
Solution
The equation of state is given by,
pV = nRT ……….. (i)
Where,
p → Pressure of gas
V → Volume of gas
n→ Number of moles of gas
R → Gas constant
T → Temperature of gas
From equation (i) we have,
`"n"/"V" = "p"/"RT"`
Replacing n with `m/M` we have
`"m"/("MV") = "p"/"RT"` ...(ii)
Where,
m → Mass of gas
M → Molar mass of gas
But `m/V = d` (d = density of gas)
Thus, from equation (ii), we have
`"d"/"M" = "p"/"RT"`
`=> "d" = ("M"/"RT")"p"`
Molar mass (M) of a gas is always constant and therefore, at constant temperature (T), `"M"/"RT"`= constant.
`"d" = ("constant") " p"`
`=> "d" prop "p"`
Hence, at a given temperature, the density (d) of gas is proportional to its pressure (p).
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