Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x^{2} \sin ^{2} x d x$ Solution: Let $I=\int x^{2} \sin ^{2} x d x$ We know that, $\sin ^{2} x=\frac{1-\cos 2 x}{2}$ $=\int x^{2}\left(\frac{1-\cos 2 x}{2}\right) d x$ Using integration by parts, $=\int \frac{x^{2}}{2} d x-\int \frac{x^{2} \cos 2 x}{2} d x$ $=\frac{x^{3}}{6}-\frac{1}{2}\left[\int x^{2} \cos 2 x d x\right]$ Using integration by parts in second integral, $=\frac{x^{3}}{6}-\frac{1}{2}\left[x^{2} \int \cos 2 x d x-\int \frac{d}{d x} ...
Read More →Mark the tick against the correct answer in the following:
Question: Mark the tick against the correct answer in the following: Let $S$ be the set of all triangles in a plane and let $R$ be a relation on $S$ defined by $\Delta_{1} S \Delta_{2} \Leftrightarrow \Delta_{1} \equiv A_{2}$. Then, $R$ is A. reflexive and symmetric but not transitive B. reflexive and transitive but not symmetric C. symmetric and transitive but not reflexive D. an equivalence relation Solution: According to the question, Given set $\mathrm{S}=\{\ldots$ All triangles in plane.......
Read More →Why can we not determine the order
Question: Why can we not determine the order of a reaction by taking into consideration the balanced chemical equation? Solution: Order is an experimental quantity. For example, in complex reactions, the rate and order are dependent on the slow step reactions, so in such cases, the order is not completely dependent on the balanced chemical equations. We cannot determine the order by taking into consideration the balanced chemical equation....
Read More →Why molecularity is applicable
Question: Why molecularity is applicable only for elementary reactions and order is applicable for elementary as well as complex reactions? Solution: Molecularity is applicable only for elementary reactions as they are the single-step reactions and the rate depends on the concentration of each molecule, whereas in case of complex reactions there are multiple reactions involved and thus molecularity holds no meaning....
Read More →Why can’t molecularity of any reaction
Question: Why cant molecularity of any reaction be equal to zero? Solution: Zero molecularity means there is no reactant, as a result of which a reaction cannot occur. So if there is a reaction then its molecularity will be greater than zero....
Read More →Why in the redox titration of KMnO4vs oxalic acid,
Question: Why in the redox titration of KMnO4vs oxalic acid, we heat oxalic acid solution before starting the titration? Solution: We heat oxalic acid solution because without heating it is a slow process as energy greater than the activation energy is required for a reaction. So to increase the energy, the temperature must be increased which is only possible by heating the oxalic acid solution....
Read More →Thermodynamic feasibility of the reaction
Question: Thermodynamic feasibility of the reaction alone cannot decide the rate of the reaction. Explain with the help of one example. Solution: Thermodynamic feasibility of the reaction alone cannot decide the rate of the reaction because for a reaction to occur it is important for the molecules to have an energy greater than the activation energy. For example, Diamond Graphite ΔG= -ve ΔG negative means that the reaction is feasible but this reaction is a slow process as energy is less than th...
Read More →Why does the rate of any reaction
Question: Why does the rate of any reaction generally decrease during the reaction? Solution: The rate of a reaction decreases because the reactants will proceed to convert into the products. The rate of a reaction always depends on the concentration of the reactants....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\log x}{x^{n}} d x$ Solution: Let $I=\int \frac{\log x}{x^{n}} d x=\int \log x \frac{1}{x^{n}} d x$ Using integration by parts, $\int \log x \frac{1}{x^{n}} d x=\log x \int \frac{1}{x^{n}} d x-\int \frac{d}{d x} \log x \int \frac{1}{x^{n}} d x$ We know that, $\int \mathrm{x}^{\mathrm{n}} \mathrm{dx}=\frac{\mathrm{x}^{\mathrm{n}+1}}{\mathrm{n}+1}$ $=\log x\left(\frac{x^{1-n}}{1-n}\right)-\int \frac{1}{x}\left(\frac{x^{1-n}}{1-n}\right) d x$ ...
Read More →Why is the probability of reaction
Question: Why is the probability of reaction with molecularity higher than three very rare? Solution: For a reaction having more than three molecules, the proper orientation is not possible that makes these reactions rare. Therefore the probability of more than three molecules colliding simultaneously will be small....
Read More →Oxygen is available in plenty in
Question: Oxygen is available in plenty in the air yet fuels do not burn by themselves at room temperature. Explain. Solution: Oxygen is available in plenty of air yet fuels do not burn by themselves at room temperature because reactants must have a minimum amount of energy known as the activation energy to form a product....
Read More →Why does the rate of a reaction increase
Question: Why does the rate of a reaction increase with a rise in temperature? Solution: Rate of reaction increases with the rise in temperature because at the higher temperature the fraction of particles collision to cross the energy barrier will be large. Therefore the rate also becomes faster....
Read More →The reaction between H2(g) and O2(g)
Question: The reaction between H2(g) and O2(g) is highly feasible yet allowing the gases to stand at room temperature in the same vessel does not lead to the formation of water. Explain. Solution: According to Maxwell Boltzmann energy distribution curve when temperature T becomes T+10 degree Celsius then the effective collision and energy of molecules increases resulting in the formation of a product....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x^{n} \log x d x$ Solution: Let $I=\int x^{n} \log x d x$ Using integration by parts, $I=\log x \int x^{n} d x-\int \frac{d}{d x} \log x \int x^{n} d x$ We know that, $\int x^{n} d x=\frac{x^{n+1}}{n+1}$ and $\frac{d}{d x} \log x=\frac{1}{x}$ $=\log x \frac{x^{n+1}}{n+1}-\int \frac{1}{x} \times \frac{x^{n+1}}{n+1} d x$ $=\log x \frac{x^{n+1}}{n+1}-\int \frac{x^{n}}{n+1} d x$ $=\log x \frac{x^{n+1}}{n+1}-\frac{1}{n+1}\left[\int x^{n} d x\right]$ W...
Read More →For a zero-order reaction will the molecularity
Question: For a zero-order reaction will the molecularity be equal to zero? Explain. Solution: The molecularity of any reaction is the number of reacting species taking part in an elementary reaction. The zero molecularity means there is no reactant so the reaction does not occur. Therefore molecularity cannot be zero for a reaction....
Read More →For a certain reaction large fraction of molecules
Question: For a certain reaction large fraction of molecules has energy more than the threshold energy, yet the rate of reaction is very slow. Why? Solution: The two most important conditions for a reaction to occur according to the collision theory are: 1. Energy is greater than activation energy and proper orientation of reactant molecules at the time of the collision. So when the reaction occurs then molecules that dont have a proper orientation decreases the rate of a reaction....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x \cos ^{2} x d x$ Solution: Let $I=\int x \cos ^{2} x d x$ Using integration by parts, $I=x \int \cos ^{2} x d x-\int \frac{d}{d x} x \int \cos ^{2} x d x$ We know that, $\cos ^{2} x=\frac{\cos 2 x+1}{2}$ $=x \int\left[\frac{\cos 2 x+1}{2}\right] d x-\int\left[1 \int\left[\frac{\cos 2 x+1}{2}\right] d x\right] d x$ We know that, $\int \cos n x=\frac{\sin n x}{n}$ $=\frac{x}{2}\left[\frac{\sin 2 x}{2}+x\right]-\frac{1}{2} \int\left(x+\frac{\sin 2...
Read More →For a reaction A + B → Products,
Question: For a reaction A + B Products, the rate law is Rate = k [A][B]3/2 Can the reaction be an elementary reaction? Explain. Solution: The reaction cannot be an elementary reaction because the order is different. For an elementary reaction, the molecularity and order should be the same....
Read More →Derive an expression to calculate
Question: Derive an expression to calculate the time required for completion of the zero-order reaction. Solution: Consider a reaction, R P t=0 Here R is the reactant and P is the product. Rate = k[R]0 t=t (Instantaneous rate) dR/dt = k , dR =-kdt On integrating both sides, ʃDr = -k ʃdt [R] = -kt + I At t = 0, [R]= [R0] which makes I= R0 [R] = [R0] kt. (i) Here [R] = concentration of reactant at time t. [R0] = initial concentration of the reactant. This reaction is known as the integrated rate e...
Read More →Mark the tick against the correct answer in the following:
Question: Mark the tick against the correct answer in the following: Let $S$ be the set of all real numbers and let $R$ be a relation on $S$, defined by $a R b \Leftrightarrow(1+a b)0$. Then, $R$ is A. reflexive and symmetric but not transitive B. reflexive and transitive but not symmetric C. symmetric and transitive but not reflexive D. none of these Solution: According to the question, Given set $S=\{\ldots \ldots,-2,-1,0,1,2 \ldots \ldots\}$ And $R=\{(a, b): a, b \in S$ and $(1+a b)0\}$ Formu...
Read More →In a reaction, if the concentration of reactant A
Question: In a reaction, if the concentration of reactant A is tripled, the rate of reaction becomes twenty-seven times. What is the order of the reaction? Solution: Consider a rate of a reaction with reactant A Rate = k[A] 27 ˟ Rate = k[A] By dividing the equations we get = 3 Therefore the order of the reaction becomes three....
Read More →For which type of reactions,
Question: For which type of reactions, order and molecularity have the same value? Solution: For an elementary reaction, the order is the same as molecularity. Elementary reactions are those reactions which occur in a single step....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x \operatorname{cosec}^{2} x d x$ Solution: Let $I=\int x \operatorname{cosec}^{2} x d x$ Using integration by parts, $I=x \int \operatorname{cosec}^{2} x d x-\int \frac{d}{d x} x \int \operatorname{cosec}^{2} x d x$ We know that, $\int \operatorname{cosec}^{2} x d x=-\cot x$ and $\int \cot x d x=\log |\sin x|$ $=x \times-\cot x-\int-\cot x d x$ $=-x \cot x+\log |\sin x|+c$...
Read More →How can you determine the rate law of
Question: How can you determine the rate law of the following reaction? 2NO (g) + O2 (g) 2NO2 (g) Solution: The rate of the reaction can be determined by Rate = k[NO]2[O2]1 It can be measured by the rate of the reaction as a function of initial concentration by keeping the concentration of one of the reactants constant and changing the other reactant....
Read More →Write the rate equation for the reaction 2A + B → C
Question: Write the rate equation for the reaction 2A + B C if the order of the reaction is zero. Solution: The rate equation for the reaction 2A + B C Rate = k[A]0[B]0 The powers of the concentration of the reactants will be equal to zero. In the zero-order reactions, the rate of the equation is equal to the rate constant....
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