To determine the equivalent value of the ideal gas constant in different units, specifically [tex]\(\frac{mmHg \cdot L}{mol \cdot K}\)[/tex], we start with the given ideal gas constant in the units [tex]\(\frac{atm \cdot L}{mol \cdot K}\)[/tex]:
[tex]\[ R = 0.08206 \frac{atm \cdot L}{mol \cdot K} \][/tex]
We need to convert this constant from its original units to the new units. We'll use the conversion factor between [tex]\(atm\)[/tex] and [tex]\(mmHg\)[/tex]. Specifically, we know that:
[tex]\[ 1 \, atm = 760 \, mmHg \][/tex]
To perform the unit conversion, we multiply the given ideal gas constant by the conversion factor from [tex]\(atm\)[/tex] to [tex]\(mmHg\)[/tex]:
[tex]\[ \text{New gas constant} = 0.08206 \times 760 \][/tex]
Now, performing the multiplication:
[tex]\[ 0.08206 \times 760 = 62.3656 \][/tex]
Thus, the equivalent value of the ideal gas constant in the units [tex]\(\frac{mmHg \cdot L}{mol \cdot K}\)[/tex] is:
[tex]\[ \boxed{62.3656 \frac{mmHg \cdot L}{mol \cdot K}} \][/tex]
