For a first-order process, the half-life is t1/2 = ln 2 / k, so it depends on the rate constant k but not on the initial amount. Temperature affects k through the Arrhenius relationship: higher temperature raises k. Therefore, as temperature increases, the half-life decreases. An example like a thermal decomposition that follows first order fits this perfectly: heating the sample speeds up the reaction (larger k), so you reach half of the reactant remaining sooner (a smaller t1/2). The other descriptions don’t align with this behavior. A second-order process has t1/2 that depends on the initial concentration and, although k increases with temperature, the stated trend of t1/2 increasing with temperature isn’t generally correct. A zero-order process would have t1/2 that changes with k and initial concentration, not a constant value, so saying it’s constant with temperature is inaccurate. And a first-order process cannot have t1/2 increasing with temperature because t1/2 is inversely related to k, which grows with temperature.

In a reversible reaction, equilibrium occurs when the forward and reverse rates balance each other. The forward rate is k_forward times the concentration of A, and the reverse rate is k_reverse times the concentration of B. At equilibrium, k_forward[A] = k_reverse[B]. Rearranging gives [B]/[A] = k_forward/k_reverse. The equilibrium constant for this 1:1 system is defined as K = [B]/[A], so K equals the ratio of the forward to reverse rate constants, and at equilibrium [B]/[A] = K.

Temperature changes affect the rate constant through the Arrhenius form k = A e^(-Ea/RT). Here Ea is the activation energy and A is the frequency factor. As temperature increases, the ratio Ea/RT decreases, making the exponent less negative. A less negative exponent means the exponential term grows, so k increases. Physically, more molecules have enough energy to overcome the activation barrier, leading to more successful collisions per unit time. Since the exponential term is always positive, k cannot become negative, and increasing temperature does not make k stay the same or decrease.

Reaction rate is the speed at which a reaction proceeds, i.e., how fast concentrations change with time. It’s defined as the change in concentration of a reactant (negative sign) or a product (positive sign) per unit time. For a simple reaction A → products, the rate can be written as rate = -d[A]/dt = d[Product]/dt. The usual units are molarity per second (M/s). Rate can be described as an average rate over a time interval or as an instantaneous rate at a particular moment. The other ideas describe different things: the product of concentrations isn’t the rate itself, the equilibrium constant tells how far the reaction goes at equilibrium, and the energy change per mole is a thermodynamic property, not the speed of the reaction.