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The value of k cannot be negative because the reaction rate cannot be negative and the relationship we know is based on the concentrations and time not of k because k is constant for every reaction at a constant pressure and temperature.
Yes, temperature will cause a change in k because an increase in temperature will increase the force of the collisions between molecules which gives more molecules enough energy to overcome the energy barrier and therefore speed up the rate of production of products. k can never be negative.
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