Question

the volume v of a given mass of a gas varies directly as the temperature t and inversely as the pressure p. if v = 231 cm³ when t = 300°k (kelvin) and p = 20 lb/cm², what is the volume when t = 504°k and p = 24 lb/cm²? v = □ (simplify your answer. type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Find the constant of variation

Since \(V\) varies directly as \(T\) and inversely as \(P\), the formula is \(V = k\frac{T}{P}\).
When \(V = 231\mathrm{cm}^3\), \(T = 300\mathrm{K}\), and \(P=20\mathrm{lb/cm}^2\).
Substitute these values into the formula: \(231=k\frac{300}{20}\).
Simplify the right - hand side: \(\frac{300}{20}=15\), so \(231 = 15k\).
Solve for \(k\): \(k=\frac{231}{15}=15.4\).

Step2: Calculate the volume

Now we know \(k = 15.4\), \(T = 504\mathrm{K}\), and \(P = 24\mathrm{lb/cm}^2\).
Substitute into \(V=k\frac{T}{P}\): \(V = 15.4\times\frac{504}{24}\).
First, calculate \(\frac{504}{24}=21\).
Then \(V=15.4\times21\).
\(V = 323.4\).

Answer:

\(323.4\)