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Question: For a gaseous reaction \[A\left( g \right)\to \text{Product}\], which one of the following is correc...

For a gaseous reaction A(g)→ProductA\left( g \right)\to \text{Product}, which one of the following is correct relation among dpdt,dndt\dfrac{dp}{dt},\dfrac{dn}{dt} and dcdt\dfrac{dc}{dt}

& \dfrac{dp}{dt}=\text{Rate of relation in atm se}{{\text{c}}^{-1}} \\\ & \dfrac{dc}{dt}=\text{Rate of relation in molarity se}{{\text{c}}^{-1}} \\\ & \dfrac{dn}{dt}=\text{ Rate of reaction in mol se}{{\text{c}}^{-1}} \\\ \end{aligned} \right)$$ A. $$\dfrac{dc}{dt}=\dfrac{dn}{dt}=\dfrac{-dp}{dt}$$ B. $$\dfrac{-dc}{dt}=\dfrac{1}{v}\dfrac{dn}{dt}=\dfrac{-1}{RT}\dfrac{dp}{dt}$$ C. $$\dfrac{dc}{dt}=\dfrac{v}{RT}\dfrac{dn}{dt}=\dfrac{dp}{dt}$$. D. None of these
Explanation

Solution

Hint: We can solve these type of question by considering the equation of state for an ideal gas which is given as PV=nRTPV=nRT
We can differentiate the equation with respect to time and find the relation between the quantities which are asked also. Also we must know that
Concentration=Number of molesVolume of gas\text{Concentration}=\dfrac{\text{Number of moles}}{\text{Volume of gas}}.

Complete answer:
From the equation of the state which is PV=nRTPV=nRT
Where P = Pressure exerted by the gas
V = Volume of gas
n = moles of gas
R = universal gas constant
T = Temperature

PV=nRTPV=nRT (differentiating the equation on both sides we get)
dp.Vdt=RT.dndt\dfrac{dp.V}{dt}=RT.\dfrac{dn}{dt}
⇒1Vdndt=1RTdpdt ........... 1\Rightarrow \dfrac{1}{V}\dfrac{dn}{dt}=\dfrac{1}{RT}\dfrac{dp}{dt}\text{ }...........\text{ }1
Now Consider, PV=nRTPV=nRT or P=nVRTP=\dfrac{n}{V}RT
Concentration =nV=C=\dfrac{n}{V}=C
Substituting this value in as one equation we get, P=CRTP=CRT
Differentiating with respect to dt on both side we get,
dpdt=dcdtRT\dfrac{dp}{dt}=\dfrac{dc}{dt}RT
dcdt=1RTdpdt\dfrac{dc}{dt}=\dfrac{1}{RT}\dfrac{dp}{dt}
Hence, the correct relationship between the given value are:
−dcdt=−1vdndt=−1RTdpdt\dfrac{-dc}{dt}=\dfrac{-1}{v}\dfrac{dn}{dt}=\dfrac{-1}{RT}\dfrac{dp}{dt}
Hence correct option in option B.

Note: In thermodynamics the equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure volume and temperature or internal energy as and when required.