To determine the initial concentration of the material, we need to use the first-order reaction formula. A first-order reaction is characterized by the equation:
[tex]\[ C_t = C_0 e^{-kt} \][/tex]
where:
- [tex]\( C_t \)[/tex] is the concentration of the material at time [tex]\( t \)[/tex],
- [tex]\( C_0 \)[/tex] is the initial concentration,
- [tex]\( k \)[/tex] is the rate constant,
- [tex]\( t \)[/tex] is the time.
Given:
- The rate constant, [tex]\( k = 5.55 \times 10^{-3} \, \text{s}^{-1} \)[/tex],
- The time, [tex]\( t = 4 \, \text{minutes} \)[/tex]. It’s important to convert this time to seconds because the rate constant is given in seconds. There are 60 seconds in a minute, so [tex]\( t = 4 \times 60 = 240 \, \text{seconds} \)[/tex],
- The concentration after 4 minutes, [tex]\( C_t = 44.1 \)[/tex] g.
We need to determine the initial concentration [tex]\( C_0 \)[/tex]. Rearrange the first-order reaction formula to solve for [tex]\( C_0 \)[/tex]:
[tex]\[ C_0 = \frac{C_t}{e^{-kt}} \][/tex]
Now substitute the known values into the equation:
[tex]\[ C_0 = \frac{44.1}{e^{- (5.55 \times 10^{-3} \times 240)}} \][/tex]
[tex]\[ C_0 = \frac{44.1}{e^{-1.332}} \][/tex]
To perform the calculation [tex]\( e^{-1.332} \approx 0.264 \)[/tex]:
[tex]\[ C_0 = \frac{44.1}{0.264} \approx 167.0778 \][/tex]
Therefore, the initial concentration is approximately 167 g. When comparing this result with the provided choices:
- 12.3 g
- 2.09 g
- 167 g
- 1.8 g
- 44.9 g
The closest match is 167 g.
Hence, the initial concentration of the material was:
[tex]\[ \boxed{167 \text{ g}} \][/tex]
