The rate of the reaction `A to` Products is given by `(dx)/(dt)=k[A]^(1//2)`. The units of the rate constants are:

Options

A. `lit mol^-1 sec^-1`
B. `lit^(1//2) mol^(-1//2) sec^-1`
C. `lit^(-1//2) mol^(1//2) sec^-1`
D. `lit^(3//2) mol^(-1//2) sec^-1`

Answer

To determine the units of the rate constant \( k \) for the reaction given by the rate equation:

\[ \frac{dx}{dt} = k [A]^{1/2} \]

we will follow these steps:

### Step 1: Identify the order of the reaction
The rate equation is expressed as:

\[ \frac{dx}{dt} = k [A]^{1/2} \]

Here, the exponent of concentration \([A]\) is \( \frac{1}{2} \). Therefore, the order of the reaction is \( \frac{1}{2} \).

### Step 2: Write the general form of the rate equation
The general form of the rate equation is:

\[ \text{Rate} = k [A]^n \]

where \( n \) is the order of the reaction. In this case, \( n = \frac{1}{2} \).

### Step 3: Determine the units of rate and concentration
The units of the rate of reaction \( \frac{dx}{dt} \) are typically expressed in terms of concentration change over time, which is:

\[ \text{Units of Rate} = \text{mol L}^{-1} \text{s}^{-1} \]

The concentration \([A]\) has units of:

\[ \text{Units of Concentration} = \text{mol L}^{-1} \]

### Step 4: Substitute the units into the rate equation
From the rate equation, we can express the units of \( k \):

\[ \text{Units of Rate} = \text{Units of } k \times \text{Units of } [A]^{1/2} \]

Substituting the known units:

\[ \text{mol L}^{-1} \text{s}^{-1} = k \times (\text{mol L}^{-1})^{1/2} \]

### Step 5: Calculate the units of \( k \)
Now, we can rearrange to find the units of \( k \):

\[ k = \frac{\text{mol L}^{-1} \text{s}^{-1}}{(\text{mol L}^{-1})^{1/2}} = \frac{\text{mol L}^{-1} \text{s}^{-1}}{\text{mol}^{1/2} \text{L}^{-1/2}} \]

This simplifies to:

\[ k = \text{mol}^{1/2} \text{L}^{1/2} \text{s}^{-1} \]

### Step 6: Finalize the units of \( k \)
Thus, the final units of the rate constant \( k \) are:

\[ \text{Units of } k = \text{mol}^{1/2} \text{L}^{-1/2} \text{s}^{-1} \]

### Conclusion
The units of the rate constant \( k \) for the reaction \( A \to \text{Products} \) with the given rate law are:

\[ \text{mol}^{1/2} \text{L}^{-1/2} \text{s}^{-1} \]