Today we are going to discuss about Combined Gas Law or sometimes it is known as the Characteristic Gas Equation. So, welcome to my YouTube channel Mechanical Engineering Management. Combined Gas Law. In actual practice, three variables that means pressure, volume and temperature change simultaneously. Therefore, Boyle's Law and Charles'Law are combined to get the relationship between pressure, volume and temperature.
So, the equation derived is known as combined gas flow. So, very simple we are going to combine boil flow and chal flow to find the relation between pressure, volume and temperature simultaneously. So, let a given mass expand from state 1 to state 2 according to boil flow and that we have discussed in earlier video. So, if I want to plot it on pressure versus volume diagram. So this is the pressure on the y-axis and volume that is always on x-axis.
Assume the state 1 over here and corresponding to that you can mention over here it is pressure P1 and volume V1. Now you know that according to Boyle's law temperature is always constant so that the process 1,2 is always curve on the pressure versus volume diagram that means on the PV diagram. So, this is a state 2, this is the process 1, 2 according to the Boile slope. You know that for the Boile slope temperature is constant and so that you can write PV is equal to constant and corresponding to state 2 I can mention over here it is P2 and volume V2 further it expands. from state 2 to state 3 according to Charles slope because we are going to combine Boyle slope and Charles slope.
So, you know that for the Charles slope pressure is always constant and so that it will be always the horizontal line on p-v diagram. So, I can assume the state 3 over here and this is the 2-3 process according to Charles slope. and corresponding to state 3 I can mention the pressure P3 over here and that is always equal to pressure P2 because of it is the horizontal line and volume V3 and so that I can say it is V by T is equal to constant because of it is the Charles law.
Now this is my earlier figure that we have already discussed. So with the help of this figure I am going to start the analysis. So, first apply Boyle's law to process 1, 2 and you know that this is what p is equal to constant. I can say for the state 1, p1, v1 is equal to p2, v2. Now, from this equation further simplification v2 is equal to p1, v1 by p2.
Let's say it is equation number 1. Now, I am going to apply the Charles law for the process 2, 3. and the relation is V by T is equal to constant. So, apply Charles law to the process 2 3 and so that I can say V2 by T2 is equal to V3 by T3. Now here you know that this is actually the constant temperature curve and so that the temperature is constant at state 1 as well as at state 2. So, that I can say T1 is equal to T2. So, now I can put the value over here t1 instead of t2. So, it is v2 by t1 is equal to v3 by t3.
Let's say it is equation number 2. Now, put the value of v2 from the equation number 1 into equation number 2. So, you know that v2 is equal to p1 v1 by p2. divided by T1 as it is is equal to as it is and V3 by T3 as it is. So, further simplification you can understand P1 V1 by T1 is equal to P2 is over this side. So, it is P2 V3 by T3.
Now, from this equation you can now imagine that here it is P1 V1 by T1 and here it is P2 V3 by T3. So, you can understand it should be P3. So, from this idea you can start the analysis. So, from the figure you can understand 2 3 process is the constant pressure process and so that you can say P2 is equal to P3 and that I have written over here. So now I can put the value over here instead of p2 I can put the p3.
So it is p1 v1 by t1 is equal to p3 v3 by t3. So now you can say in generalize that each and every state always pv by t is constant. So here it is for the state 1 and here it is for the state 3. So in generalize I can say PV by T is always constant and that constant is always R that is actually characteristic gas constant. Let's say it is equation number 3. So, so far we have assumed the gas that we have taken for the analysis that is 1 kg. Now it is for the m kg of mass then I can say PV by T is equal to MR. Let's say it is equation number 4. Next from this equation I can say PV is equal to MRT and this is the equation number 5 and which is very important equation and that equation number 5 is known as characteristic gas equation.
In the previous slide that PV by T is equal to R that means equation number 3 is known as combined gas law. Now look at this R constant. R is called as the characteristic gas constant.
Its value is different for the different gas. Its value for the air is 0.287 kilojoule per kg kelvin. And that you have to remember because of in numerical portion sometimes this value is not given to you. So at that time you can take for the air R is equal to 0.287 kilojoule per kg kelvin.
the product of molecular mass of gas and its specific gas constant is same for all gases. That means m into r that is always constant for all gases and this constant is known as universal gas constant Ru. So, I can say mathematically Ru is equal to m into r.
So, keep in mind that r is the different for the different gas. But your Ru that means universal gas constant is always constant for all the gases and that is the product of molecular mass into characteristic gas constant and that is always 8.314 kilojoule per kg mole Kelvin. So thanks my dear friends.
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