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How many times the rate of reaction increases at 200C for a reaction having the activation energies in the presence and absence of catalyst as 50 kJ/mol and 75 kJ/mol?
- a)1000
- b)10000
- c)30000
- d)50000
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
How many times the rate of reaction increases at 200C for a reaction h...
The rate of reaction will increase 28,592 times, i.e. 30,000 times.
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How many times the rate of reaction increases at 200C for a reaction h...
Given:
Activation energy with catalyst (Ec) = 50 kJ/mol
Activation energy without catalyst (Enc) = 75 kJ/mol
Formula:
The rate constant (k) of a reaction is given by the Arrhenius equation:
k = A * e^(-Ea/RT)
where:
k = rate constant
A = pre-exponential factor
Ea = activation energy
R = gas constant
T = temperature
Analysis:
The rate constant is directly proportional to the rate of reaction.
Case 1: With catalyst (Ec)
Let's assume the rate constant with catalyst is kc.
Case 2: Without catalyst (Enc)
Let's assume the rate constant without catalyst is knc.
Calculation:
To compare the rate of reaction at two different temperatures, we can use the ratio of rate constants.
At 200°C:
Temperature (T) = 200 + 273 = 473 K
Case 1: With catalyst (Ec)
kc = A * e^(-Ec/RT)
Case 2: Without catalyst (Enc)
knc = A * e^(-Enc/RT)
Taking the ratio of rate constants:
kc/knc = [A * e^(-Ec/RT)] / [A * e^(-Enc/RT)]
The pre-exponential factor (A) cancels out, so we are left with:
kc/knc = e^(-Ec/RT) / e^(-Enc/RT)
Since the temperatures (T) and the gas constant (R) are the same in both cases, we can simplify the equation further:
kc/knc = e^(-Ec/RT + Enc/RT)
Using the properties of exponents, we can rewrite the equation as:
kc/knc = e^[-(Ec - Enc)/RT]
Substituting the given activation energies:
kc/knc = e^[-(50 - 75)/(8.314 * 473)]
Simplifying the exponent:
kc/knc = e^(25/8.314 * 473)
Calculating the value:
kc/knc ≈ 29999
Therefore, the rate of reaction increases approximately 30,000 times at 200°C.
Activation energy with catalyst (Ec) = 50 kJ/mol
Activation energy without catalyst (Enc) = 75 kJ/mol
Formula:
The rate constant (k) of a reaction is given by the Arrhenius equation:
k = A * e^(-Ea/RT)
where:
k = rate constant
A = pre-exponential factor
Ea = activation energy
R = gas constant
T = temperature
Analysis:
The rate constant is directly proportional to the rate of reaction.
Case 1: With catalyst (Ec)
Let's assume the rate constant with catalyst is kc.
Case 2: Without catalyst (Enc)
Let's assume the rate constant without catalyst is knc.
Calculation:
To compare the rate of reaction at two different temperatures, we can use the ratio of rate constants.
At 200°C:
Temperature (T) = 200 + 273 = 473 K
Case 1: With catalyst (Ec)
kc = A * e^(-Ec/RT)
Case 2: Without catalyst (Enc)
knc = A * e^(-Enc/RT)
Taking the ratio of rate constants:
kc/knc = [A * e^(-Ec/RT)] / [A * e^(-Enc/RT)]
The pre-exponential factor (A) cancels out, so we are left with:
kc/knc = e^(-Ec/RT) / e^(-Enc/RT)
Since the temperatures (T) and the gas constant (R) are the same in both cases, we can simplify the equation further:
kc/knc = e^(-Ec/RT + Enc/RT)
Using the properties of exponents, we can rewrite the equation as:
kc/knc = e^[-(Ec - Enc)/RT]
Substituting the given activation energies:
kc/knc = e^[-(50 - 75)/(8.314 * 473)]
Simplifying the exponent:
kc/knc = e^(25/8.314 * 473)
Calculating the value:
kc/knc ≈ 29999
Therefore, the rate of reaction increases approximately 30,000 times at 200°C.
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How many times the rate of reaction increases at 200C for a reaction having the activation energies in the presence and absence of catalyst as 50 kJ/mol and 75 kJ/mol?a)1000b)10000c)30000d)50000Correct answer is option 'C'. Can you explain this answer?
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