While non-permeable solutes require a carrier protein to allow passage into a cell, they do not require ATP or energy if they are traveling down their concentration gradient; energy is only required if they are traveling against their gradient.

All other options are examples of cellular functions that require ATP usage.  

Nonpolar molecules and very small polar molecules can freely pass through the lipid bilayer, while large, polar molecules and ions need to be aided by transport proteins. Sodium and potassium are both charged ions that would not be able to cross the membrane. Glucose and citrate are too large, and also contain polar regions.

Carbon dioxide is the only answer choice that is both small and nonpolar enough to simply diffuse across the membrane.

Transport of molecules along their gradients does not require an input of energy, while transporting molecules against their gradients requires cellular energy. Facilitated diffusion refers to the transport of a molecules along its gradient through a protein channel medium. The molecule cannot passively diffuse, usually because of size or polarity, but can still be transported without use of energy.

Active transport of any kind, including the sodium-potassium pump and any ATPases, will require energy to transport a molecule against its natural gradient.

Remember that plasma membrane receptors are found in several types of cells.

G protein coupled receptors are a class of receptors that have a G protein attached to the intracellular side. Upon ligand binding, the G protein dissociates from the receptor and binds to subsequent ion channels or effector proteins to initiate a signal cascade.

Membrane receptors are also found in B-cells, which are part of the immune system. Surfaces of B-cells contain several B-cell receptors with antibodies embedded into each receptor. These antibodies are very specific and bind foreign antigens. Binding of an antigen initiates an immune response that eventually leads to the destruction of the antigen. 

Recall that membrane receptors span the membrane; therefore, the membrane-spanning region of a receptor must contain hydrophobic residues. Hydrophobic residues are amino acids with nonpolar side chains.

In signal transduction, a ligand binds to the extracellular side of the plasma membrane receptor. This initiates a cellular response that is facilitated by the intracellular side. The intracellular region can activate a G protein, bind to an effector, or initiate other cellular responses. These responses often result in a signal cascade that affects transcription factors and alters gene expression.

The pores formed are, according to the question, sodium selective. So it is unlikely that potassium concentration changes will be a major contributor to membrane potential changes. Since sodium is postively charged, and the ions entering are sodium, the inside of the cell will become more positively charged as sodium permeability goes up. We know that sodium will enter and potassium will leave due to the established gradients determined by sodium-potassium ATPase.

Membrane potential is the difference between the electric potential inside the cell and the electric potential outside the cell. At rest, the membrane potential of most cells (including nerve cells) is between -70mV and -80mV due to the concentration of intracellular and extracellular potassium and sodium ions. The expulsion of sodium ions, in particular, contributes to positive charges outside the cell and lowers the charge inside.

Membrane potential is calculated by subtracting the potential outside the cell from the potential inside the cell.

A neuron usually has a negative resting membrane potential because the inside of the cell is more negative than the outside of the cell. This difference in polarity results from the uneven movement of sodium and potassium ions by the sodium-potassium pump. The amount of sodium ions pumped out of the cell (three) is higher than the amount of potassium ions pumped into the cell (two). The net export of positive ions contributes to the negative resting membrane potential.

Remember that depolarization is the first step of an action potential, during which the membrane potential increases rapidly. This rapid increase is attributed to the movement of sodium ions into the cell. Since sodium ions are positively charged, the inside of the cell becomes more positive and the outside of the cell becomes more negative. This change in polarity causes an increase in membrane potential. When a neuron is excited past the threshold stimulus, an action potential ensues. An action potential causes huge alterations in membrane potential due to the movement of ions during depolarization, repolarization, and hyperpolarization. The sodium-potassium pump constantly moves sodium and potassium ions against their concentration gradients, which helps maintain the negative resting membrane potential. 

During resting membrane potential, a sodium ion inside the cell will not have a higher electrical potential energy than a sodium ion outside the cell. Recall that the electrical potential energy is higher when the electrical potential is higher. The resting membrane potential is negative; therefore, the inside of the cell has a lower potential than the outside of the cell. Since the potential inside the cell is lower, a sodium ion inside the cell will have a lower potential than a sodium ion outside the cell.

Two main factors are required for a membrane potential. First, there must be a concentration or electrochemical gradient of ions. A potential difference occurs between two regions in space that have an uneven distribution of charges; therefore, for a membrane potential there must be a concentration gradient of ions (a difference between the ion concentrations) between the outside and the inside of the cell. Second, there must be a semipermeable membrane that separates the interior of the cell from the exterior. If a semipermeable membrane isn’t present, then the ions can undergo simple diffusion, which will equilibrate the concentration of ions. This would diminish the concentration gradient and the potential difference would become zero. 

Neurotransmitters are not required for establishing a membrane potential. They are chemicals that bind to receptors on membranes and initiate a cellular response.