Question

a gas is cooled from 365 k to 285 k while its volume changes from 12.8 l to 9.9 l. the initial pressure of the gas is 1.9 atm. what is the final pressure of the gas, rounded to the nearest tenth? 1.3 atm 2.9 atm 1.9 atm 9.9 atm

Explanation:

Step1: Recall Combined Gas Law

The combined gas law is \(\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}\), where \(P_1 = 1.9\) atm, \(V_1 = 12.8\) L, \(T_1 = 365\) K, \(V_2 = 9.9\) L, \(T_2 = 285\) K, and we need to find \(P_2\).

Step2: Rearrange Formula for \(P_2\)

Rearrange the combined gas law to solve for \(P_2\): \(P_2=\frac{P_1V_1T_2}{T_1V_2}\)

Step3: Substitute Values

Substitute the given values into the formula: \(P_2=\frac{1.9\times12.8\times285}{365\times9.9}\)
First, calculate the numerator: \(1.9\times12.8 = 24.32\); \(24.32\times285 = 24.32\times200+24.32\times85 = 4864 + 2067.2 = 6931.2\)
Then, calculate the denominator: \(365\times9.9 = 365\times(10 - 0.1)=3650 - 36.5 = 3613.5\)
Now, divide numerator by denominator: \(P_2=\frac{6931.2}{3613.5}\approx1.93\approx1.9\)? Wait, no, wait, let's recalculate. Wait, maybe I made a mistake. Wait, \(1.9\times12.8 = 24.32\); \(24.32\times285\): 24.32285. Let's do 24285 = 6840, 0.32285 = 91.2, so total 6840 + 91.2 = 6931.2. Denominator: 3659.9. 36510 = 3650, minus 3650.1 = 36.5, so 3650 - 36.5 = 3613.5. Then 6931.2 / 3613.5 ≈ 1.92. Wait, but the options have 1.9? Wait, maybe I miscalculated. Wait, let's check again. Wait, maybe the initial values: Wait, the volume changes from 12.8 L to 9.9 L, temperature from 365 K to 285 K. Wait, maybe I mixed up V1 and V2? No, P1V1/T1 = P2V2/T2, so P2 = (P1V1T2)/(T1V2). Wait, maybe my calculation is wrong. Let's use a calculator approach:

1.9 12.8 = 24.32; 24.32 285 = 24.32 285. Let's compute 24 285 = 6840, 0.32 * 285 = 91.2, so total 6931.2.

Denominator: 365 9.9 = 365(10 - 0.1) = 3650 - 36.5 = 3613.5.

Now, 6931.2 / 3613.5 ≈ 1.92, which rounds to 1.9? Wait, but the options have 1.9 as an option. Wait, but let's check again. Wait, maybe I made a mistake in the formula. Wait, combined gas law: (P1V1)/T1 = (P2V2)/T2. So P2 = (P1V1T2)/(T1V2). Let's plug in the numbers again:

P1 = 1.9, V1 = 12.8, T1 = 365; V2 = 9.9, T2 = 285.

So P2 = (1.9 12.8 285) / (365 * 9.9)

Calculate numerator: 1.912.8 = 24.32; 24.32285 = 24.32285. Let's do 28524 = 6840, 285*0.32 = 91.2, so 6840+91.2=6931.2

Denominator: 3659.9=365(10-0.1)=3650-36.5=3613.5

Now, 6931.2 / 3613.5 ≈ 1.92, which is approximately 1.9 atm. Wait, but let's check the options. The options are 1.3, 2.9, 1.9, 9.9. So 1.9 is an option. Wait, but maybe my calculation is wrong. Wait, let's use another approach. Let's compute (1.9 12.8)/365 = (24.32)/365 ≈ 0.0666. Then multiply by 285: 0.0666285 ≈ 19.0. Then divide by 9.9: 19.0 / 9.9 ≈ 1.92, which is approximately 1.9. So the final pressure is approximately 1.9 atm.

Answer:

1.9 atm (Option: 1.9 atm)