For the reaction : `2A + B rarr C + D`, measurement of the rate of the reaction at varying concentrations are given below:
`{:("Trial No.",[A],[B],"Rate in mol L"^(-1)s^(-1)),(1,0.010,0.010,2.5),(2,0.010,0.020,5.0),(3,0.030,0.020,45.0):}`
The overall order of the reaction is :

Options

A. 3
B. 2
C. 1
D. 0

Explanation

To determine the overall order of the reaction \(2A + B \rightarrow C + D\), we will analyze the given data from the trials and derive the rate law expression.

### Step 1: Write the Rate Law Expression
The rate law expression for the reaction can be written as:
\[ \text{Rate} = k [A]^x [B]^y \]
where \(x\) is the order with respect to \(A\) and \(y\) is the order with respect to \(B\).

### Step 2: Analyze Trial 1
From Trial 1:
- \([A] = 0.010\) M
- \([B] = 0.010\) M
- Rate = 2.5 mol L\(^{-1}\) s\(^{-1}\)

Substituting into the rate law:
\[ 2.5 = k (0.010)^x (0.010)^y \tag{1} \]

### Step 3: Analyze Trial 2
From Trial 2:
- \([A] = 0.010\) M
- \([B] = 0.020\) M
- Rate = 5.0 mol L\(^{-1}\) s\(^{-1}\)

Substituting into the rate law:
\[ 5.0 = k (0.010)^x (0.020)^y \tag{2} \]

### Step 4: Divide Equation (1) by Equation (2)
Dividing (1) by (2):
\[ \frac{2.5}{5.0} = \frac{k (0.010)^x (0.010)^y}{k (0.010)^x (0.020)^y} \]
This simplifies to:
\[ 0.5 = \frac{(0.010)^y}{(0.020)^y} \]
\[ 0.5 = \left(\frac{0.010}{0.020}\right)^y = \left(\frac{1}{2}\right)^y \]
From this, we can deduce:
\[ y = 1 \]
Thus, the order with respect to \(B\) is 1.

### Step 5: Analyze Trial 3
From Trial 3:
- \([A] = 0.030\) M
- \([B] = 0.020\) M
- Rate = 45.0 mol L\(^{-1}\) s\(^{-1}\)

Substituting into the rate law:
\[ 45.0 = k (0.030)^x (0.020)^y \tag{3} \]

### Step 6: Divide Equation (2) by Equation (3)
Now, we will divide (2) by (3):
\[ \frac{5.0}{45.0} = \frac{k (0.010)^x (0.020)^y}{k (0.030)^x (0.020)^y} \]
This simplifies to:
\[ \frac{1}{9} = \frac{(0.010)^x}{(0.030)^x} \]
\[ \frac{1}{9} = \left(\frac{0.010}{0.030}\right)^x = \left(\frac{1}{3}\right)^x \]
From this, we can deduce:
\[ x = 2 \]
Thus, the order with respect to \(A\) is 2.

### Step 7: Calculate Overall Order
The overall order of the reaction is the sum of the individual orders:
\[ \text{Overall Order} = x + y = 2 + 1 = 3 \]

### Final Answer
The overall order of the reaction is **3**.

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