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Quantum-dot cellular automata (QCA) offers a new paradigm for molecular electronics, a paradigm in which information transmission and processing depend on electrostatic interactions between charges in arrays of cells composed of quantum dots. Fundamental questions about the operational temperature and functional gain of devices built from molecular-scale QCA cells are addressed in this paper through a statistical-mechanical model based on electrostatic interactions. The model provides exact solutions for the thermodynamic constraints on operation of small arrays of cells (up to 15). An Ising approximation dramatically reduces the computational task and allows modeling of the thermodynamic behavior of semi-infinite QCA wires. The probability of getting the correct output from a QCA device for a given input depends on temperature, cell size, cell-cell distance, effective dielectric constant of the medium, and the number of cells in the array. Using parameters derived from molecular candidates for QCA cells, the statistical-mechanical model predicts that majority gates should give correct output at temperatures of up to 450 K, while wires of thousands to millions of QCA cells are predicted to operate as functional devices at room temperature.