Question

For a gaseous reaction \[A\left( g \right)\to \text{Product}\], which one of the following is correct relation among \[\dfrac{dp}{dt},\dfrac{dn}{dt}\] and \[\dfrac{dc}{dt}\]
\[\left( \begin{align}
  & \dfrac{dp}{dt}=\text{Rate of relation in atm se}{{\text{c}}^{-1}} \\
 & \dfrac{dc}{dt}=\text{Rate of relation in molarity se}{{\text{c}}^{-1}} \\
 & \dfrac{dn}{dt}=\text{ Rate of reaction in mol se}{{\text{c}}^{-1}} \\
\end{align} \right)\]
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

Answer 1

Hint: We can solve these type of question by considering the equation of state for an ideal gas which is given as \[PV=nRT\]
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
\[\text{Concentration}=\dfrac{\text{Number of moles}}{\text{Volume of gas}}\].

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

\[PV=nRT\] (differentiating the equation on both sides we get)
\[\dfrac{dp.V}{dt}=RT.\dfrac{dn}{dt}\]
\[\Rightarrow \dfrac{1}{V}\dfrac{dn}{dt}=\dfrac{1}{RT}\dfrac{dp}{dt}\text{ }...........\text{ }1\]
Now Consider, \[PV=nRT\] or \[P=\dfrac{n}{V}RT\]
Concentration \[=\dfrac{n}{V}=C\]
Substituting this value in as one equation we get, \[P=CRT\]
Differentiating with respect to dt on both side we get,
\[\dfrac{dp}{dt}=\dfrac{dc}{dt}RT\]
\[\dfrac{dc}{dt}=\dfrac{1}{RT}\dfrac{dp}{dt}\]
Hence, the correct relationship between the given value are:
\[\dfrac{-dc}{dt}=\dfrac{-1}{v}\dfrac{dn}{dt}=\dfrac{-1}{RT}\dfrac{dp}{dt}\]
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.