The National Institute of Science and Technology (NIST) kinetics database lists the rate constant, k, of a particular gaseous reaction as 5.78× 10–10 cm3·molecule^–1·s^–1 at 298 K. Convert this constant, k, to torr^-1 X S^-1.
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To convert the rate constant, k, from cm^3·molecule^–1·s^–1 to torr^-1 X s^-1, you need to consider the unit conversions for volume and pressure.
Step 1: Convert cm^3 to L
1 cm^3 = 1 × 10^−3 L
Step 2: Convert from L·molecule^–1·s^–1 to L·molecule^–1·min^–1
1 s = 60 s
Step 3: Convert from L·molecule^–1·min^–1 to L·molecule^–1·s^–1
1 min = 60 s
Step 4: Convert from L·molecule^–1·s^–1 to torr^-1 X s^-1
1 torr = 1/760 atm
1 atm = 101325 Pa
1 Pa = 1 N·m^–2
1 N = 1 kg·m·s^–2
Now let's perform the conversions step-by-step:
Step 1: Convert cm^3 to L
1 cm^3 = 1 × 10^−3 L
Therefore, 5.78 × 10^–10 cm^3·molecule^–1·s^–1 = 5.78 × 10^–13 L·molecule^–1·s^–1
Step 2: Convert from L·molecule^–1·s^–1 to L·molecule^–1·min^–1
1 s = 60 s
Therefore, 5.78 × 10^–13 L·molecule^–1·s^–1 = 5.78 × 10^–13 L·molecule^–1·min^–1
Step 3: Convert from L·molecule^–1·min^–1 to L·molecule^–1·s^–1
1 min = 60 s
Therefore, 5.78 × 10^–13 L·molecule^–1·min^–1 = 9.63 × 10^–15 L·molecule^–1·s^–1
Step 4: Convert from L·molecule^–1·s^–1 to torr^-1 X s^-1
1 torr = 1/760 atm
1 atm = 101325 Pa
1 Pa = 1 N·m^–2
1 N = 1 kg·m·s^–2
And 101325 Pa = 1 × 10^5 N·m^–2
Therefore, to convert from L·molecule^–1·s^–1 to torr^-1 X s^-1, we need to divide the value by (1 × 10^5) / 760.
9.63 × 10^–15 L·molecule^–1·s^–1 / [(1 × 10^5) / 760] ≈ 1.16 × 10^–19 torr^-1·s^-1
Thus, the rate constant, k, for the gaseous reaction is approximately 1.16 × 10^–19 torr^-1 X s^-1.
To convert the rate constant, k, from cm^3·molecule^-1·s^-1 to torr^-1·s^-1, we need to use the ideal gas law, which relates pressure (P), volume (V), temperature (T), and number of molecules (n) of a gas.
The ideal gas law formula is: PV = nRT
Where:
P = pressure (in torr)
V = volume (in liters)
n = number of molecules
R = ideal gas constant (0.0821 L·atm·K^-1·mol^-1)
T = temperature (in Kelvin)
Since we are converting from cm^3 to liters, we need to divide the rate constant, k, by 1000 to convert it to liters.
k (in cm^3·molecule^-1·s^-1) = 5.78 × 10^-10 cm^3·molecule^-1·s^-1
k (in liters·molecule^-1·s^-1) = 5.78 × 10^-10 cm^3·molecule^-1·s^-1 / 1000 = 5.78 × 10^-13 liters·molecule^-1·s^-1
Now let's solve for the pressure, P, using the ideal gas law:
PV = nRT
Since we are interested in converting the rate constant to torr^-1·s^-1, we can rewrite the equation as:
P (in torr) = (nRT) / V
We know that the number of molecules (n) is 1, the gas constant (R) is 0.0821 L·atm·K^-1·mol^-1, and the volume (V) is 1 liter.
P (in torr) = (1 molecule) × (0.0821 L·atm·K^-1·mol^-1) × (298 K) / (1 liter)
P (in torr) = 24.4518 torr
Finally, we can calculate the conversion:
k (in torr^-1·s^-1) = k (in liters·molecule^-1·s^-1) / P (in torr)
k (in torr^-1·s^-1) = (5.78 × 10^-13 liters·molecule^-1·s^-1) / (24.4518 torr)
k (in torr^-1·s^-1) = 2.37 × 10^-14 torr^-1·s^-1
Therefore, the rate constant, k, of the gaseous reaction is 2.37 × 10^-14 torr^-1·s^-1 at 298 K.