The volume of a gas is directly proportional to temperature and inversely proportional to pressure. If the gas occupies 400 cubic feet volume at some pressure, then by what percentage does the pressure need to be increased so as to compress the gas into a 4 cubic feet tank at constant temperature?a)9600%b)9700%c)9800%d)9900%Correct answer is option 'D'. Can you explain this answer?

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The volume of a gas is directly proportional to temperature and inversely proportional to pressure. If the gas occupies 400 cubic feet volume at some pressure, then by what percentage does the pressure need to be increased so as to compress the gas into a 4 cubic feet tank at constant temperature?

a)
9600%
b)
9700%
c)
9800%
d)
9900%

Correct answer is option 'D'. Can you explain this answer?

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Most Upvoted Answer

The volume of a gas is directly proportional to temperature and invers...

V α T/P
Since temperature is constant, therefore
V α T/P
⇒ V₁P₁ = V₂P₂
400P₁ = 4P₂
P₂ = 100P₁
So, percentage increase

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The volume of a gas is directly proportional to temperature and invers...

Given:
Volume of gas at initial pressure (V1) = 400 cubic feet
Volume of gas at final pressure (V2) = 4 cubic feet
Percentage increase in pressure = ?

Solution:
We are given that the volume of a gas is directly proportional to temperature and inversely proportional to pressure.

So, we can write the equation as:
V ∝ T/P

Where V is the volume, T is the temperature, and P is the pressure.

Step 1: Find the initial pressure (P1) using the initial volume (V1) and the given equation.

V1 ∝ T/P1
(V1 * P1) = k (constant)

Step 2: Find the final pressure (P2) using the final volume (V2) and the given equation.

V2 ∝ T/P2
(V2 * P2) = k (constant)

Step 3: Find the percentage increase in pressure.

The percentage increase in pressure can be calculated using the formula:

Percentage increase = [(new value - old value) / old value] * 100

In this case:
Old value = P1
New value = P2

Substituting the values into the formula:
Percentage increase = [(P2 - P1) / P1] * 100

Step 4: Calculate the percentage increase in pressure.

We know that (V1 * P1) = (V2 * P2)

Substituting the given values:
(400 * P1) = (4 * P2)

Simplifying the equation:
P1 = (4 * P2) / 400
P1 = P2 / 100

Now, substituting this value of P1 in the percentage increase formula, we get:
Percentage increase = [(P2 - (P2 / 100)) / (P2 / 100)] * 100
Percentage increase = [99P2 / 100] * 100
Percentage increase = 9900%

Therefore, the pressure needs to be increased by 9900% in order to compress the gas into a 4 cubic feet tank at constant temperature.

Hence, the correct answer is option D) 9900%.

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