Math Problem Statement

The volume of a gas V held at a constant temperature in a closed container varies inversely with its pressure P. If the volume of a gas is 900 cubic centimeters left parenthesis cm cubed right parenthesis when the pressure is 200 millimeters of mercury​ (mm Hg), find the volume when the pressure is 400 mm Hg. Question content area bottom Part 1 When the pressure is 400 mm​ Hg, the volume is    enter your response here ▼ mm Hg per cm cubed . mm Hg. cm cubed . cm cubed per mm Hg.

Solution

The relationship between the volume VV and pressure PP of a gas at a constant temperature follows Boyle's Law, which states that volume and pressure are inversely proportional. This can be written as:

V1PV \propto \frac{1}{P}

or, equivalently,

VP=kV \cdot P = k

where kk is a constant.

Step 1: Find the constant kk

We are given that the volume V=900cm3V = 900 \, \text{cm}^3 when the pressure P=200mm HgP = 200 \, \text{mm Hg}. Using the equation VP=kV \cdot P = k:

900200=k900 \cdot 200 = k

k=180,000k = 180,000

Step 2: Use the constant kk to find the volume at P=400mm HgP = 400 \, \text{mm Hg}

Now, using the same equation VP=kV \cdot P = k, we substitute P=400mm HgP = 400 \, \text{mm Hg} and k=180,000k = 180,000:

V400=180,000V \cdot 400 = 180,000

Solving for VV:

V=180,000400=450cm3V = \frac{180,000}{400} = 450 \, \text{cm}^3

Final Answer:

When the pressure is 400mm Hg400 \, \text{mm Hg}, the volume is 450 cubic centimeters (cm³).

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Related Questions:

  1. What happens to the volume if the pressure doubles again to 800 mm Hg?
  2. Can this law be applied to non-ideal gases? If so, how?
  3. How does temperature affect the relationship between pressure and volume in gases?
  4. What are the conditions for Boyle's Law to hold true?
  5. What would happen to the volume if the pressure is reduced to 100 mm Hg?

Tip:

In inverse relationships like this, as one variable increases, the other decreases proportionally, making the constant very useful for calculations!

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Math Problem Analysis

Mathematical Concepts

Inverse Proportions
Gas Laws
Algebra

Formulas

V ∝ 1/P
V * P = k (Boyle's Law)

Theorems

Boyle's Law

Suitable Grade Level

Grades 9-12