Certainly! Let's walk through the process of solving for the constant [tex]\( k \)[/tex] in the ideal gas law equation [tex]\( PV = kT \)[/tex] with given values for pressure [tex]\( P \)[/tex], volume [tex]\( V \)[/tex], and temperature [tex]\( T \)[/tex].
### Step-by-Step Solution:
1. Understand the Ideal Gas Law:
The ideal gas law states that the product of pressure ([tex]\( P \)[/tex]) and volume ([tex]\( V \)[/tex]) of a gas is proportional to its temperature ([tex]\( T \)[/tex]), with [tex]\( k \)[/tex] being the proportionality constant:
[tex]\[
PV = kT
\][/tex]
Our goal is to solve for [tex]\( k \)[/tex].
2. Identify Given Values:
We are given specific values for pressure ([tex]\( P \)[/tex]), volume ([tex]\( V \)[/tex]), and temperature ([tex]\( T \)[/tex]):
[tex]\[
P = 2 \text{ atm (atmospheres)}
\][/tex]
[tex]\[
V = 3 \text{ liters}
\][/tex]
[tex]\[
T = 300 \text{ Kelvin}
\][/tex]
3. Rearrange the Ideal Gas Law to Solve for [tex]\( k \)[/tex]:
To isolate [tex]\( k \)[/tex], we can rearrange the equation:
[tex]\[
k = \frac{PV}{T}
\][/tex]
4. Substitute the Given Values into the Equation:
Let's substitute [tex]\( P = 2 \)[/tex] atm, [tex]\( V = 3 \)[/tex] liters, and [tex]\( T = 300 \)[/tex] Kelvin into the equation:
[tex]\[
k = \frac{(2 \text{ atm}) \times (3 \text{ liters})}{300 \text{ K}}
\][/tex]
5. Perform the Calculation:
Now, calculate the value:
[tex]\[
k = \frac{6}{300}
\][/tex]
[tex]\[
k = 0.02
\][/tex]
6. Summary of the Result:
Thus, the constant [tex]\( k \)[/tex] that satisfies the ideal gas law for the given values of pressure, volume, and temperature is:
[tex]\[
k = 0.02 \text{ atm·L}/\text{K}
\][/tex]
In conclusion, using the given values for pressure, volume, and temperature in the ideal gas law equation, we determined that the constant [tex]\( k \)[/tex] is [tex]\( 0.02 \text{ atm·L}/\text{K} \)[/tex].
