To solve the given equation [tex]\(\frac{P_1 V_1}{n_1 T_1} = \frac{P_2 V_2}{n_2 T_3}\)[/tex] for [tex]\(n_1\)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[
\frac{P_1 V_1}{n_1 T_1} = \frac{P_2 V_2}{n_2 T_3}
\][/tex]
2. To isolate [tex]\(n_1\)[/tex], first multiply both sides of the equation by [tex]\(n_1 T_1\)[/tex]:
[tex]\[
P_1 V_1 = \frac{P_2 V_2 \cdot n_1 T_1}{n_2 T_3}
\][/tex]
3. Then, multiply both sides of the equation by [tex]\(\frac{n_2 T_3}{P_2 V_2}\)[/tex]:
[tex]\[
n_1 T_1 = \frac{P_1 V_1 \cdot n_2 T_3}{P_2 V_2}
\][/tex]
4. Finally, divide both sides by [tex]\(T_1\)[/tex] to solve for [tex]\(n_1\)[/tex]:
[tex]\[
n_1 = \frac{P_1 V_1 \cdot n_2 T_3}{P_2 V_2 \cdot T_1}
\][/tex]
Now we compare this derived expression with the provided options:
- Option 1: [tex]\(n_1 = \frac{P_2 V_2 T_2 n_2}{P_1 V_1 T_1}\)[/tex]
- Option 2: [tex]\(n_1 = \frac{P_2 V_2 T_1}{P_1 V_1 n_2 T_3}\)[/tex]
- Option 3: [tex]\(n_1 = \frac{P_2 V_2 T_2}{P_1 V_1 T_1 n_2}\)[/tex]
- Option 4: [tex]\(n_1 = \frac{P_1 V_1 n_2 T_3}{P_2 V_2 T_1}\)[/tex]
The correct solution we derived matches exactly with Option 4:
[tex]\[
n_1 = \frac{P_1 V_1 n_2 T_3}{P_2 V_2 T_1}
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{4}
\][/tex]
