3 moles of a gaseous mixture having volume V and temperature T^' are compressed to (1 / 5) th of

3 moles of a gaseous mixture having volume V and temperature...

3 moles of a gaseous mixture having volume $V$ and temperature $T^{\prime}$ are compressed to $(1 / 5)$ th of its initial volume. The change in its adiabatic compressibility, if gas obeys the equation $P V^{19 / 13}=$ constant, is: $(R=8.3 \mathrm{~J} / \mathrm{mol}-\mathrm{K})$ (a) $\Delta C=-0.0248 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}$ (b) $\Delta C=-0.035 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}$ (c) $\Delta C=\cdots 0.0426 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}$ (d) $\Delta C=-0.0137 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}$

3 moles of a gaseous mixture having volume $V$ and temperature $T^{\prime}$ are compressed to $(1 / 5)$ th of its initial volume. The change in its adiabatic compressibility, if gas obeys the equation $P V^{19 / 13}=$ constant, is:
$(R=8.3 \mathrm{~J} / \mathrm{mol}-\mathrm{K})$
(a) $\Delta C=-0.0248 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}$
(b) $\Delta C=-0.035 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}$
(c) $\Delta C=\cdots 0.0426 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}$
(d) $\Delta C=-0.0137 \frac{V}{T} \mathrm{~m}^{2} / \mathrm{N}$

Key Concepts

Ideal Gas Law

The ideal gas law, given by PV = nRT, is a fundamental relation that connects pressure, volume, and temperature for an ideal gas. It provides the basis for understanding how changes in state variables are interrelated in gaseous systems and is essential when analyzing thermodynamic processes, including compression and expansion of gases.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings, meaning all changes in the system’s internal energy are due solely to work done on or by the system. This concept is essential in the study of thermodynamics because it simplifies the analysis of rapid processes or insulated systems, where the heat transfer is negligible.

Polytropic Process

A polytropic process is characterized by the relation PV^n = constant, where the exponent n can vary depending on the process conditions. This general equation encompasses many specific cases including adiabatic processes (when n equals the ratio of specific heats) and is used to model various reversible processes in thermodynamics.

Adiabatic Compressibility

Adiabatic compressibility is a measure of the change in volume of a system under an applied pressure change when the process is adiabatic. It is defined by the derivative of volume with respect to pressure, usually considering the constraints imposed by the adiabatic condition. This parameter is crucial for understanding how a gas responds to compression or expansion without the influence of heat exchange.

Transcript

00:01 Hello students in this question we have a gases mixture having number of moles equals to 3 and volume v and temperature t this is compressed to the volume v dash equals to 1 by 5 times of volume v okay and the in the adiabatic compressibility the gas obvious the law pv to the power 19 divided by 13 equals to constant okay, so we have to calculate the change in the adiabatic compressibility that is delta c or delta k okay so this is the equation of the gas which is following so this can be compared with pv -r -rish -to -the -power gamma equals to constant so from here we get adiabetic coefficient is equals to 19 divided by 30 okay now we can calculate here that we can calculate we have the volume ratio okay so we can calculate the pressure ratio so we can write that p v to the the power gamma this is equals to p -dash v -to -the -power gamma okay so we can write that p and v to the power gamma this is equals to p -dash and v -dash which is equals to this v by five to the power this gamma okay so we can compare this equation so we will obtain that p -dash is equal to five to the power gamma multiplied the pressure p okay now we can write that the compressibility c is given by 1 by beta and beta is the bulk modulus and this is equals to 1 by gamma multiplied by p okay so delta c this will be equals to the 1 by gamma and 1 by pressure p dash minus 1 by p so 1 by gamma and 1 by p dash this will be equals to the 1 by 5 to the power gamma and modpled by this pressure p and minus 1 by pressure p so delta c from here will be equals to the 1 by gamma pressure p and 1 by 5 to the power gamma minus 1.

02:09 And this pressure p can be written from the ideal gas equation that is pv equals to nrt.

02:16 So delta c will be equals to the volume v divided by 3 r t made plied by gamma and 1 by 5 to the power gamma minus 1.

02:27 Now we can substitute all the values.

02:30 We have all the values, except this volume and temperature...

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