Calculate the partial pressure of monatomic hydrogen in hydrogen gas at 2000 K and 1 atm pressure.
Given:
(i) For ½ H2 -> H, DeltaHo298=217,990 J; DeltaSo298=49.35 J/K.
(ii) Assume that the heat capacity of monatomic gas to be 3/2R
(iii)The heat capacity of H2 assume to be 31 J/(mol K)
Answer: PH = 5.22x10-4 atm
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-mats2008
To calculate the partial pressure of monatomic hydrogen (H) in hydrogen gas (H2) at 2000 K and 1 atm pressure, we can use the following steps:
1. Use the van't Hoff equation to relate the equilibrium constant (K) to the change in standard Gibbs free energy (ΔG) with temperature:
ΔG = ΔH - TΔS
where ΔH is the change in enthalpy, ΔS is the change in entropy, and T is the temperature in Kelvin.
2. Calculate the change in standard Gibbs free energy (ΔG) at 2000 K using ΔH and ΔS:
ΔG = ΔHo298 + (ΔSo298 * (T - 298))
3. Convert the value of ΔG from joules to moles (since ΔG has units of J/mol) by dividing by the gas constant (R = 8.314 J/(mol K)):
ΔG /= -R
4. Use the equation relating equilibrium constant (K) to ΔG:
K = e^(ΔG/RT)
where e is the base of the natural logarithm.
5. Calculate the partial pressure of monatomic hydrogen (PH) using the definition of K for the dissociation of H2:
K = (PH)^2 / P(H2)
Rearrange the equation to solve for PH:
PH = sqrt(K * P(H2))
6. Plug in the given values:
- ΔHo298 = 217,990 J
- ΔSo298 = 49.35 J/K
- Heat capacity of monatomic gas (Cv) = 3/2R
- Heat capacity of H2 (Cp) = 31 J/(mol K)
- P(H2) = 1 atm
- T = 2000 K
Now, let's calculate the partial pressure of monatomic hydrogen (PH):
Step 1: Calculate ΔG
ΔG = ΔHo298 + (ΔSo298 * (T - 298))
ΔG = 217,990 J + (49.35 J/K * (2000 K - 298))
ΔG = 217,990 J + (49.35 J/K * 1702 K)
ΔG = 217,990 J + 83,927.7 J
ΔG = 301,917.7 J
Step 2: Convert ΔG to moles
ΔG /= -R = -301,917.7 J / (-8.314 J/(mol K))
ΔG = 36,335.39 mol
Step 3: Calculate K
K = e^(ΔG/RT)
K = exp(36,335.39 mol / (8.314 J/(mol K) * 2000 K))
K ≈ 6.963 x 10^108
Step 4: Calculate PH
PH = sqrt(K * P(H2))
PH = sqrt(6.963 x 10^108 * 1 atm)
PH ≈ 5.22 x 10^-4 atm
Therefore, the partial pressure of monatomic hydrogen in hydrogen gas at 2000 K and 1 atm pressure is approximately 5.22 x 10^-4 atm.