Question

a gas has an initial volume of 168 cm³ at a temperature of 255 k and a pressure of 1.6 atm. the pressure of the gas decreases to 1.3 atm, and the temperature of the gas increases to 285 k. what is the final volume of the gas? 122 cm³ 153 cm³ 185 cm³ 231 cm³

Explanation:

Step1: Recall Combined Gas Law

The combined gas law is \(\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}\), where \(P_1 = 1.6\) atm, \(V_1 = 168\) \(cm^3\), \(T_1 = 255\) K, \(P_2 = 1.3\) atm, \(T_2 = 285\) K, and we need to find \(V_2\).

Step2: Rearrange Formula for \(V_2\)

Rearrange the combined gas law to solve for \(V_2\): \(V_2=\frac{P_1V_1T_2}{P_2T_1}\)

Step3: Substitute Values

Substitute the given values into the formula: \(V_2=\frac{1.6\times168\times285}{1.3\times255}\)

First, calculate the numerator: \(1.6\times168 = 268.8\); \(268.8\times285 = 268.8\times200+268.8\times85 = 53760 + 22848 = 76608\)

Then, calculate the denominator: \(1.3\times255 = 331.5\)

Now, divide numerator by denominator: \(V_2=\frac{76608}{331.5}\approx231\) (Wait, no, wait, let's recalculate. Wait, maybe I made a mistake. Wait, let's do it step by step again.

Wait, \(P_1 = 1.6\), \(V_1 = 168\), \(T_2 = 285\); \(P_2 = 1.3\), \(T_1 = 255\)

So \(V_2=\frac{1.6\times168\times285}{1.3\times255}\)

Calculate \(1.6\times168 = 268.8\)

\(268.8\times285\): Let's compute 268.8285. 268.8200=53760, 268.880=21504, 268.85=1344. So 53760+21504=75264+1344=76608

Denominator: 1.3*255=331.5

Now, 76608 / 331.5. Let's divide numerator and denominator by 1.5: 76608/1.5=51072; 331.5/1.5=221. Wait, no, better to do decimal division. 331.5230=331.5200 + 331.5*30=66300 + 9945=76245. 76608 - 76245=363. 363/331.5≈1.1. So total≈230 + 1.1≈231.16, which is approximately 231 \(cm^3\). Wait, but let's check the options. The last option is 231 \(cm^3\). Wait, but maybe I messed up the formula? Wait, combined gas law: \(\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}\), so \(V_2 = V_1\times\frac{P_1}{P_2}\times\frac{T_2}{T_1}\)

So \(V_1 = 168\), \(\frac{P_1}{P_2}=\frac{1.6}{1.3}\approx1.2308\), \(\frac{T_2}{T_1}=\frac{285}{255}\approx1.1176\)

Multiply them together: 168 1.2308 1.1176. First, 1681.2308≈1681.2 + 1680.0308≈201.6 + 5.1744≈206.7744. Then, 206.77441.1176≈206.77441 + 206.77440.1176≈206.7744 + 24.316≈231.09, which is approximately 231 \(cm^3\).

Answer:

231 \(cm^3\) (The option is 231 \(cm^3\))