Which of the following graphs correctly represents the variation of β=-((dV/dP)) (1/V) with P for an ideal gas at constant temperature?

Question

Which of the following graphs correctly represents the variation of β=-((dV/dP)) (1/V) with P for an ideal gas at constant temperature? (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

For an ideal gas, the differential from of gas law is

V d P + P d V = n R d T

At constant temperature

d T = 0

\Rightarrow

V d P + P d V = 0

\frac{- d V}{d P} = (\frac{V}{P}) or - (\frac{d V}{d P}) \frac{1}{V} = \frac{1}{P} \Rightarrow β = \frac{1}{P}

βP = co n s t an t

Hence the graph could be a rectangular hyperbola as shown in (a).

Q. Which of the following graphs correctly represents the variation of β=−(dPdV​)V1​ with P for an ideal gas at constant temperature?

For an ideal gas, the differential from of gas law is VdP+PdV=nRdT At constant temperature dT=0 ⇒ VdP+PdV=0 dP−dV​=(PV​)or−(dPdV​)V1​=P1​⇒β=P1​ βP=constant Hence the graph could be a rectangular hyperbola as shown in (a).

Q. Which of the following graphs correctly represents the variation of $β = - (\frac{d V}{d P}) \frac{1}{V}$ with $P$ for an ideal gas at constant temperature?