Question Video: Determining the Relationship between the Pressure and Number of Moles of an Ideal Gas | Nagwa Question Video: Determining the Relationship between the Pressure and Number of Moles of an Ideal Gas | Nagwa

Question Video: Determining the Relationship between the Pressure and Number of Moles of an Ideal Gas Physics

For an ideal gas where the volume and temperature are held constant, which of the following is the correct proportionality relation between the pressure, 𝑃, of the gas, and the number of moles, 𝑛? [A] 𝑃 ∝ (1/𝑛) [B] 𝑃 ∝ (1/𝑛²) [C] 𝑃 ∝ 𝑛² [D] 𝑃 ∝ 𝑛 [E] 𝑃 ∝ βˆšπ‘›

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Video Transcript

For an ideal gas where the volume and temperature are held constant, which of the following is the correct proportionality relation between the pressure, 𝑃, of the gas and the number of moles, 𝑛? (A) 𝑃 is proportional to one divided by 𝑛. (B) 𝑃 is proportional to one divided by 𝑛 squared. (C) 𝑃 is proportional to 𝑛 squared. (D) 𝑃 is proportional to 𝑛. (E) 𝑃 is proportional to the square root of 𝑛.

We’re told that we’re working in this scenario with an ideal gas. This gas can be described then by the ideal gas law. This law says that for an ideal gas, the gas’s pressure times its volume is equal to the number of moles of the gas multiplied by a constant times the gas’s temperature. We’re interested in the relationship between the pressure, 𝑃, and the number of moles of the gas, 𝑛. To make that relationship clearer, let’s divide both sides of this equation by the volume of the gas, 𝑉, which cancels that factor out on the left. This gives us an expression where the pressure, 𝑃, is the subject.

On the right side of this equation, the symbol 𝑅 represents a constant. And we’re told in our problem statement that the gas volume and temperature are held constant. Therefore, 𝑅 times 𝑇 divided by 𝑉 is all equal to a constant value. If we wanted to, we could give a name to this value. Say we call it capital 𝐢. In that case, our ideal gas law equation can be written as 𝑃, the pressure, is equal to a constant, 𝐢, multiplied by the number of moles of gas, 𝑛.

A mathematically equivalent way to write this equation is to say that 𝑃 is proportional to 𝑛. This is true because once we have 𝑛, we just multiply it by a constant value to get the pressure, 𝑃. 𝑃 is proportional to 𝑛. And we see that this agrees with one of our answer options. Answer choice (D) correctly says that the proportionality relation between the pressure 𝑃 of the gas and the number of moles 𝑛 of the gas is that 𝑃 is directly proportional to 𝑛.

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