Highly stable monodispersed nano Cu hydrosols were facilely prepared by an aqueous chemical reduction method through selecting copper hydroxide (Cu(OH)₂) as the copper precursor, poly(acrylic acid) (PAA) and ethanol amine (EA) as the complexing agents, and hydrazine hydrate as the reducing agent. The size of the obtained Cu colloidal nanoparticles was controlled from 0.96 to 26.26 nm by adjusting the dosage of the copper precursor. Moreover, the highly stable nano Cu hydrosols could be easily concentrated and re-dispersed in water meanwhile maintaining good dispersibility. A model catalytic reaction of reducing p-nitrophenol with NaBH₄ in the presence of nano Cu hydrosols with different sizes was performed to set up the relationship between the apparent kinetic rate constant (k_app) and the particle size of Cu catalysts. The experimental results indicate that the corresponding k_app showed an obvious size-dependency. Calculations revealed that k_app was directly proportional to the surface area of Cu catalyst nanoparticles, and also proportional to the reciprocal of the particle size based on the same mass of Cu catalysts. This relationship might be a universal principle for predicting and assessing the catalytic efficiency of Cu nanoparticles. The activation energy (E_a) of this catalytic reaction when using 0.96 nm Cu hydrosol as a catalyst was calculated to be 9.37 kJ mol⁻¹, which is considered an extremely low potential barrier. In addition, the synthesized nano Cu hydrosols showed size-dependent antibacterial activities against Pseudomonas aeruginosa (P. aeruginosa) and the minimal inhibitory concentration of the optimal sample was lower than 5.82 μg L⁻¹.