Solution: To find the rate of formation of \( NO_2 \), we first need to determine the change in concentration of \( N_2O_4 \) over the given time period.
Initial concentration of \( N_2O_4 \): \([N_2O_4]_0 = 3 \, \text{mol L}^{-1}\)
Final concentration after 30 minutes: \([N_2O_4] = 2.75 \, \text{mol L}^{-1}\)
The change in concentration of \( N_2O_4 \) over 30 minutes is:
\[ \Delta [N_2O_4] = [N_2O_4]_0 - [N_2O_4] = 3 - 2.75 = 0.25 \, \text{mol L}^{-1} \]
According to the reaction:
\[ 2N_2O_4 \rightarrow 4NO_2 \]
For every 2 moles of \( N_2O_4 \) that decompose, 4 moles of \( NO_2 \) are formed, so the ratio is:
\[ \frac{4 \, \text{mol} \, NO_2}{2 \, \text{mol} \, N_2O_4} = 2 \]
The change in concentration of \( NO_2 \) formed is:
\[ \Delta [NO_2] = 2 \times \Delta [N_2O_4] = 2 \times 0.25 = 0.50 \, \text{mol L}^{-1} \]
The rate of formation of \( NO_2 \) over 30 minutes is:
\[ \text{Rate} = \frac{\Delta [NO_2]}{\Delta t} = \frac{0.50 \, \text{mol L}^{-1}}{30 \, \text{min}} = \frac{0.50}{30} \, \text{mol L}^{-1} \text{min}^{-1} = \frac{1}{60} \, \text{mol L}^{-1} \text{min}^{-1} \]
Given \( x \times 10^{-3} = \frac{1}{60} \), we can find \( x \):
\[ x = \frac{1}{60} \times 1000 = 16.67 \approx 17 \]
Thus, the value of \( x \) is: 17
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32