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 $P_1, V_1, T_1$ are initial pressure, volume, temperature; $P_2, V_2, T_2$ are final.

Step2: Identify Given Values

$P_1 = 1.6\ \text{atm}$, $V_1 = 168\ \text{cm}^3$, $T_1 = 255\ \text{K}$; $P_2 = 1.3\ \text{atm}$, $T_2 = 285\ \text{K}$. Solve for $V_2$.

Step3: Rearrange Formula for $V_2$

$V_2 = \frac{P_1V_1T_2}{P_2T_1}$

Step4: Substitute Values

$V_2 = \frac{1.6 \times 168 \times 285}{1.3 \times 255}$
Calculate numerator: $1.6\times168 = 268.8$; $268.8\times285 = 76608$
Denominator: $1.3\times255 = 331.5$
$V_2 = \frac{76608}{331.5} \approx 231\ \text{cm}^3$? Wait, no, wait: Wait, let's recalculate. Wait, 1.6168=268.8; 268.8285: 268.8200=53760, 268.880=21504, 268.85=1344; total 53760+21504=75264+1344=76608. Denominator:1.3255=331.5. 76608/331.5: let's divide 76608 by 331.5. 331.5230=331.5200=66300, 331.5*30=9945; 66300+9945=76245. 76608-76245=363. 363/331.5≈1.095. So total≈231.095, so ~231 cm³? Wait but wait, maybe miscalculation. Wait no, wait the options: 231 is an option. Wait but let's check again. Wait, maybe I messed up. Wait, combined gas law: $\frac{P1V1}{T1} = \frac{P2V2}{T2}$. So $V2 = (P1V1T2)/(P2T1)$. Let's compute again:

$P1=1.6$, $V1=168$, $T2=285$; $P2=1.3$, $T1=255$.

So (1.6 168 285) / (1.3 * 255)

1.6/1.3 ≈1.2308; 168285=168(200+80+5)=33600+13440+840=47880; 478801.2308≈478801.2=57456, 47880*0.0308≈1474.7; total≈58930.7; divide by 255: 58930.7/255≈231. So yes, 231 cm³. Wait but let's check the options. The last option is 231 cm³. Wait but maybe I made a mistake. Wait, no, let's do exact calculation:

1.6168 = 268.8; 268.8285 = 268.8(255 + 30) = 268.8255 + 268.830. 268.8255: 268.8200=53760, 268.850=13440, 268.85=1344; 53760+13440=67200+1344=68544. 268.830=8064. Total numerator: 68544+8064=76608. Denominator:1.3*255=331.5. 76608 ÷ 331.5: divide numerator and denominator by 1.5: 76608/1.5=51072; 331.5/1.5=221. So 51072/221≈231.09. So yes, ~231 cm³.

Answer:

231 cm³ (the option with 231 cm³)