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Question: How much should the pressure be increased in order to decrease the volume of the gas by 5% at a cons...

How much should the pressure be increased in order to decrease the volume of the gas by 5% at a constant temperature?

Explanation

Solution

When a gas expands its pressure decreases and its volume increases. This means that the pressure of the gas is inversely proportional to the volume of the gas. P1V1=P2V2{{P}_{1}}{{V}_{1}}={{P}_{2}}{{V}_{2}} can be used to solve this problem.

Complete step by step answer:
According to the question the pressure is increasing and volume is decreasing at a constant temperature, this corresponds to Boyle's law. It states that the pressure and volume of the gas are inversely proportional to each other.
It can be formulated as:
P∝1VP\propto \dfrac{1}{V}
PV=KPV=K
Where K is constant.
This relation can be converted into:
P1V1=P2V2{{P}_{1}}{{V}_{1}}={{P}_{2}}{{V}_{2}}

So according to the question V1{{V}_{1}} is V and V2{{V}_{2}} is 5% less than the original volume at a constant temperature.
For V2{{V}_{2}} can be written as:
V2=(100−5){{V}_{2}}=(100-5)%{{V}_{1}}
V2=95{{V}_{2}}=95%\text{ }{{V}_{1}}
V2=0.95 V1{{V}_{2}}=0.95\text{ }{{V}_{1}}
And we have to find the increase in pressure:
Putting the values of both volumes in the equation, we get:
P1V1=P2 0.95 V1{{P}_{1}}{{V}_{1}}={{P}_{2}}\text{ 0}\text{.95 }{{V}_{1}}
From this we get the relation of pressure:
P1=P2 x 0.95{{P}_{1}}={{P}_{2}}\text{ x 0}\text{.95}
So P1{{P}_{1}} is equal to 0.95 times the P2{{P}_{2}}

This can also be written as:
P2=P10.95=1.0526P1{{P}_{2}}=\dfrac{{{P}_{1}}}{0.95}=1.0526{{P}_{1}}
Or we can say that P2{{P}_{2}} is equal to 1.0526 times the P1{{P}_{1}}
We know that change in pressure is equal to the difference in the final pressure and the initial pressure.
=P2−P1={{P}_{2}}-{{P}_{1}}

We know the value of P2{{P}_{2}}, so putting the value of P2{{P}_{2}} in this equation we get:
=1.526P1−P1=0.0526P1=1.526{{P}_{1}}-{{P}_{1}}=0.0526{{P}_{1}}
On converting this into percentage, we get
0.0526P1 x 100 = 5.26 !!0.0526{{P}_{1}}\text{ x 100 = 5}\text{.26 }\\!\\!%\\!\\!\text{ }{{\text{P}}_{1}}
So, 5.26% of pressure must be increased in order to decrease the volume of the gas by 5%.

Note: The formula used in this equation only if the condition is specified at a constant temperature. Because this formula is based on Boyle's law.