In order to solve the Knapsack Problem with a single Continuous variable (KPC) efficiently, a novel S-shape transfer function based on Gauss error function was proposed, and a new approach of transforming a real vector into a 0-1 vector by using the proposed transfer function was given, thereby a New Binary Particle Swarm Optimization algorithm (NBPSO) was proposed. Then, based on the second mathematical model of KPC and the combination of NBPSO and the effective algorithm to deal with the infeasible solutions of KPC, a new approach to solve KPC was proposed. For validating the performance of NBPSO in solving KPS, NBPSO was utilized to solve four kinds of large-scale KPC instances, and the obtained calculation results were compared with those of Binary Particle Swarm Optimization algorithms (BPSOs) based on other S and V-shape transfer functions, Single-population Binary Differential Evolution with Hybrid encoding (S-HBDE), Bi-population Binary Differential Evolution with Hybrid encoding (B-HBDE) and Binary Particle Swarm Optimization algorithm (BPSO). The comparison results show that NBPSO is superior to the comparison algorithms in average calculation result and stability, illustrating that NBPSO has the performance better than other algorithms.