Question

a gas has an initial volume of 24.6 l at a pressure of 1.90 atm and a temperature of 335 k. the pressure of the gas increases to 3.50 atm, and the volume of the gas increases to 31.3 l. what is the final temperature of the gas? 143 k 231 k 485 k 785 k

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 ones. We need to solve for \(T_2\), so rearrange the formula: \(T_2=\frac{P_2V_2T_1}{P_1V_1}\).

Step2: Identify Given Values

\(P_1 = 1.90\) atm, \(V_1 = 24.6\) L, \(T_1 = 335\) K, \(P_2 = 3.50\) atm, \(V_2 = 31.3\) L.

Step3: Substitute Values into Formula

Substitute into \(T_2=\frac{3.50\times31.3\times335}{1.90\times24.6}\). First calculate numerator: \(3.50\times31.3 = 109.55\); \(109.55\times335 = 36699.25\). Denominator: \(1.90\times24.6 = 46.74\). Then \(T_2=\frac{36699.25}{46.74}\approx785\)? Wait, no, wait, maybe miscalculation. Wait, let's recalculate: \(3.50\times31.3 = 109.55\); \(109.55\times335\): \(109.55\times300 = 32865\), \(109.55\times35 = 3834.25\), total \(32865 + 3834.25 = 36699.25\). Denominator: \(1.90\times24.6\): \(20\times24.6 = 492\), minus \(1\times24.6 = 24.6\), so \(492 - 24.6 = 46.74\). Then \(36699.25\div46.74\approx785\)? Wait, but let's check again. Wait, maybe I messed up the numbers. Wait, initial pressure 1.90, final 3.50; initial volume 24.6, final 31.3; initial temp 335. Wait, maybe the correct calculation: \(T_2=\frac{3.50\times31.3\times335}{1.90\times24.6}\). Let's compute step by step:

\(3.50\div1.90\approx1.8421\); \(31.3\div24.6\approx1.2724\); then multiply these two: \(1.8421\times1.2724\approx2.344\); then multiply by 335: \(2.344\times335\approx785\) K. Wait, but the options include 785 K. Wait, but let me check again. Wait, maybe I made a mistake in the problem's numbers? Wait, the initial pressure is 1.90, final 3.50 (increase), initial volume 24.6, final 31.3 (increase), so temperature should increase. Let's recalculate:

\(T_2 = \frac{P_2 V_2 T_1}{P_1 V_1} = \frac{3.50 \text{ atm} \times 31.3 \text{ L} \times 335 \text{ K}}{1.90 \text{ atm} \times 24.6 \text{ L}}\)

Calculate numerator: \(3.50 \times 31.3 = 109.55\); \(109.55 \times 335 = 109.55 \times 300 + 109.55 \times 35 = 32865 + 3834.25 = 36699.25\)

Denominator: \(1.90 \times 24.6 = 46.74\)

Then \(T_2 = 36699.25 / 46.74 ≈ 785\) K. Wait, but the option is 785 K. Wait, but maybe I had a miscalculation earlier? Wait, no, let's check with another approach. Let's use exact fractions:

\(T_2 = (3.50 \times 31.3 \times 335) / (1.90 \times 24.6)\)

3.50 is 7/2, 1.90 is 19/10, 31.3 is 313/10, 24.6 is 123/5.

So substituting:

\(T_2 = \frac{(7/2) \times (313/10) \times 335}{(19/10) \times (123/5)}\)

Simplify denominators and numerators:

The 10 in numerator and denominator cancels. So:

\(T_2 = \frac{7 \times 313 \times 335 \times 5}{2 \times 19 \times 123}\)

Calculate numerator: 7313=2191; 2191335=2191(300+35)=2191300=657300, 219135=76685, total 657300+76685=733985; 7339855=3669925.

Denominator: 219=38; 38123=4674.

Then 3669925 / 4674 ≈ 785 (since 4674785=4674700=3271800, 4674*85=397290; 3271800+397290=3669090, close to 3669925, difference is 835, so approx 785 + 835/4674≈785.18, so ~785 K.

Answer:

785 K (the option with 785 K)